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Triangle Properties and Angle Theorems

46 questions

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Question 1Question →

The measures of the three interior angles of a triangle are in the ratio 2:3:52:3:5. What is the measure of the largest angle in the triangle?

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Answer: 90∘90^\circ

Answer

The correct answer is 90∘90^\circ.
Since the interior angles of a triangle sum to 180∘180^\circ and their ratio is 2:3:52:3:5, the sum of the ratio parts is 2+3+5=102 + 3 + 5 = 10. Dividing the total 180∘180^\circ by 10 parts yields 18∘18^\circ per ratio unit. The largest angle corresponds to the largest part of the ratio, which is 5. Therefore, the measure of the largest angle is 5×18∘=90∘5 \times 18^\circ = 90^\circ.

Step-by-Step Solution

1
Find the total number of parts in the ratio by adding the terms together.
The total number of parts is 2+3+5=102 + 3 + 5 = 10 parts.
This determines how many equal units the total angle measure is divided into.
2
Divide the total sum of the interior angles of a triangle by the total number of parts to find the degree measure of one part.
The sum of the interior angles of a triangle is 180∘180^\circ. Dividing by the total parts gives 180∘10=18∘\frac{180^\circ}{10} = 18^\circ per part.
The Triangle Angle Sum Theorem states that the interior angles of a triangle always sum to 180∘180^\circ.
3
Multiply the value of one part by the ratio term representing the largest angle.
The largest angle is represented by the term 5, so its measure is 5×18∘=90∘5 \times 18^\circ = 90^\circ.
This gives the measure of the largest angle in the triangle.

Key Concept

The interior angles of a triangle always sum to 180∘180^\circ, and individual angle measures can be found from a given ratio by dividing the total degrees by the sum of the ratio parts.
Question 2Question →

In △ABC\triangle ABC, the measure of exterior angle ∠BCD\angle BCD is 115∘115^\circ. If the measure of interior angle ∠A\angle A is 45∘45^\circ, what is the measure, in degrees, of interior angle ∠B\angle B?

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Answer: 70

Answer

The measure of interior angle ∠B\angle B is 7070 degrees.
By the Exterior Angle Theorem, the measure of exterior angle ∠BCD\angle BCD is equal to the sum of the two remote interior angles, ∠A\angle A and ∠B\angle B. We can write this relationship as m∠BCD=m∠A+m∠Bm\angle BCD = m\angle A + m\angle B. Substituting 115∘115^\circ for m∠BCDm\angle BCD and 45∘45^\circ for m∠Am\angle A gives 115=45+m∠B115 = 45 + m\angle B. Solving for m∠Bm\angle B yields 70∘70^\circ.

Step-by-Step Solution

1
Set up the equation using the Exterior Angle Theorem.
m∠BCD=m∠A+m∠Bm\angle BCD = m\angle A + m\angle B
The measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles.
2
Substitute the given measurements into the equation.
115=45+m∠B115 = 45 + m\angle B
The exterior angle ∠BCD\angle BCD measures 115∘115^\circ and the remote interior angle ∠A\angle A measures 45∘45^\circ.
3
Solve for the unknown angle measure by subtraction.
m∠B=70m\angle B = 70
Subtracting 4545 from both sides isolates m∠Bm\angle B.

Key Concept

Exterior Angle Theorem
Question 3Question →

A triangular framework is being constructed using three metal rods. Two of the rods have lengths of 1414 inches and 2525 inches. The third rod must have an integer length of xx inches. What is the number of possible integer values for xx?

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Answer: 27

Answer

27
The correct answer is the value representing 27 possible integers. By the Triangle Inequality Theorem, the third side xx must satisfy the inequality 25−14<x<25+1425 - 14 < x < 25 + 14, which simplifies to 11<x<3911 < x < 39. Since xx must be an integer, it can take any value from 1212 to 3838, inclusive. Counting these integers yields 38−12+1=2738 - 12 + 1 = 27 possible values.

Step-by-Step Solution

1
Apply the Triangle Inequality Theorem to set up the inequality for the third side xx.
25−14<x<25+1425 - 14 < x < 25 + 14
The Triangle Inequality Theorem states that the length of any side of a triangle must be strictly greater than the difference of the other two sides and strictly less than their sum.
2
Simplify the compound inequality to find the bounds for xx.
11<x<3911 < x < 39
Subtracting and adding the side lengths gives the range of possible values for the third side.
3
Identify the set of integers that satisfy the simplified inequality.
x∈{12,13,14,…,37,38}x \in \{12, 13, 14, \dots, 37, 38\}
Since xx must be an integer and the inequalities are strict, the minimum integer value is 1212 and the maximum is 3838.
4
Calculate the count of integers in the range from 1212 to 3838, inclusive.
38−12+1=2738 - 12 + 1 = 27
To find the number of integers in an inclusive range [a,b][a, b], use the formula b−a+1b - a + 1.

Key Concept

Triangle Inequality Theorem

Alternative Method

To find the number of integers strictly between two integers aa and bb (where a<ba < b), you can use the formula (b−a)−1(b - a) - 1. For the bounds 11<x<3911 < x < 39, the calculation is (39−11)−1=28−1=27(39 - 11) - 1 = 28 - 1 = 27.
Estimated Time:1m 15s
Question 4Question →

In △XYZ\triangle XYZ, the measure of ∠X\angle X is 50∘50^\circ, and the measure of ∠Y\angle Y is twice the measure of ∠X\angle X. What is the measure of ∠Z\angle Z?

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Answer: 30∘30^\circ

Answer

The correct answer is 30∘30^\circ.
Since the measure of ∠X\angle X is 50∘50^\circ and the measure of ∠Y\angle Y is twice that of ∠X\angle X, the measure of ∠Y\angle Y is 2×50∘=100∘2 \times 50^\circ = 100^\circ. Because the sum of the interior angles of any triangle is 180∘180^\circ, we can find the measure of ∠Z\angle Z by subtracting the sum of the measures of ∠X\angle X and ∠Y\angle Y from 180∘180^\circ: 180∘−(50∘+100∘)=30∘180^\circ - (50^\circ + 100^\circ) = 30^\circ. Thus, the correct option is the one stating 30∘30^\circ.

Step-by-Step Solution

1
Calculate the measure of ∠Y\angle Y using the given relationship with ∠X\angle X.
∠Y=2×50∘=100∘\angle Y = 2 \times 50^\circ = 100^\circ
The problem states that the measure of ∠Y\angle Y is twice the measure of ∠X\angle X, which is given as 50∘50^\circ.
2
Calculate the measure of ∠Z\angle Z using the Triangle Angle Sum Theorem.
∠Z=180∘−(50∘+100∘)=30∘\angle Z = 180^\circ - (50^\circ + 100^\circ) = 30^\circ
The sum of the interior angles of any triangle is always 180∘180^\circ.

Key Concept

Triangle Angle Sum Theorem
Estimated Time:45s
Question 5Question →

The measures of the three interior angles of a triangle are in the ratio 2:3:52:3:5. What is the measure, in degrees, of the largest angle of the triangle?

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Answer: 90∘90^\circ

Answer

90 degrees
The sum of the measures of the interior angles of a triangle is 180∘180^\circ. Given the ratio 2:3:52:3:5, the angles can be expressed as 2x2x, 3x3x, and 5x5x. Adding these yields 2x+3x+5x=1802x + 3x + 5x = 180, which simplifies to 10x=18010x = 180. Solving for xx gives x=18x = 18. The largest angle is represented by 5x5x. Substituting 1818 for xx gives 5×18=905 \times 18 = 90. Therefore, the measure of the largest angle is 90∘90^\circ.

Step-by-Step Solution

1
Set up an equation representing the sum of the angles in a triangle.
2x+3x+5x=1802x + 3x + 5x = 180
The sum of the measures of the interior angles of any triangle is always 180∘180^\circ. We can represent the angles as 2x2x, 3x3x, and 5x5x using the given ratio.
2
Solve for the scale factor xx.
10x=180⇒x=1810x = 180 \Rightarrow x = 18
Combine like terms to find the total number of parts, then divide 180180 by 1010 to find the value of one part.
3
Calculate the measure of the largest angle.
5×18=905 \times 18 = 90
The largest angle corresponds to the largest term in the ratio, which is 5x5x. Multiplying the scale factor 1818 by 55 gives the measure of the largest angle.

Key Concept

The interior angles of a triangle sum to 180∘180^\circ. Ratios can be solved by defining a common multiplier for each part and setting their sum equal to the total.
Estimated Time:45s
Question 6Question →

In △PQR\triangle PQR, the measure of ∠P\angle P is 50∘50^\circ, and the measure of the exterior angle at vertex QQ is 110∘110^\circ. If the bisector of ∠PRQ\angle PRQ intersects side PQPQ at point SS, what is the measure, in degrees, of ∠PRS\angle PRS?

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Answer: 30

Answer

The measure of ∠PRS\angle PRS is 30∘30^\circ.
By using the linear pair relationship, the interior angle ∠PQR\angle PQR is found to be 70∘70^\circ. Applying the triangle angle sum theorem, the third interior angle ∠PRQ\angle PRQ is 180∘−(50∘+70∘)=60∘180^\circ - (50^\circ + 70^\circ) = 60^\circ. The angle bisector RSRS divides this angle into two equal parts, resulting in a measure of 30∘30^\circ for ∠PRS\angle PRS.

Step-by-Step Solution

1
Find the interior angle at vertex QQ
∠PQR=70∘\angle PQR = 70^\circ
The interior and exterior angles at a vertex are supplementary, summing to 180∘180^\circ.
2
Find the measure of interior angle ∠PRQ\angle PRQ
∠PRQ=60∘\angle PRQ = 60^\circ
The sum of the interior angles in any triangle is 180∘180^\circ.
3
Calculate the measure of the bisected angle ∠PRS\angle PRS
∠PRS=30∘\angle PRS = 30^\circ
An angle bisector divides the angle into two equal measures.

Key Concept

Triangle Angle Sum Theorem and Exterior Angle Relationships
Estimated Time:1m 30s
Question 7Question →

In △PQR\triangle PQR, the lengths of sides PQPQ and PRPR are equal. If the measure of ∠P\angle P is 70∘70^\circ, what is the measure, in degrees, of ∠Q\angle Q?

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Answer: 55

Answer

The measure of ∠Q\angle Q is 55∘55^\circ.
Since PQ=PRPQ = PR, △PQR\triangle PQR is an isosceles triangle with base QRQR, which means the base angles ∠Q\angle Q and ∠R\angle R have equal measures. The sum of the interior angles of a triangle is 180∘180^\circ. Setting up the equation gives 70∘+2(∠Q)=180∘70^\circ + 2(\angle Q) = 180^\circ. Subtracting 70∘70^\circ from both sides yields 2(∠Q)=110∘2(\angle Q) = 110^\circ. Dividing by 2, we find that the measure of ∠Q\angle Q is 55∘55^\circ.

Step-by-Step Solution

1
Identify the properties of the given triangle.
△PQR\triangle PQR is an isosceles triangle with base QRQR and base angles ∠Q=∠R\angle Q = \angle R.
A triangle with two equal sides is isosceles, and the angles opposite those sides are equal in measure.
2
Apply the triangle angle sum theorem.
∠P+∠Q+∠R=180∘\angle P + \angle Q + \angle R = 180^\circ
The sum of the measures of the interior angles of any triangle is always 180∘180^\circ.
3
Substitute the known values and solve for ∠Q\angle Q.
70∘+2(∠Q)=180∘  ⟹  2(∠Q)=110∘  ⟹  ∠Q=55∘70^\circ + 2(\angle Q) = 180^\circ \implies 2(\angle Q) = 110^\circ \implies \angle Q = 55^\circ
Substituting ∠P=70∘\angle P = 70^\circ and ∠R=∠Q\angle R = \angle Q allows us to solve the linear equation for the unknown angle measure.

Key Concept

Isosceles Triangle Properties and Triangle Angle Sum Theorem
Estimated Time:45s
Question 8Question →

In quadrilateral ABCDABCD, diagonal ACAC divides the figure into two triangles, △ABC\triangle ABC and △ACD\triangle ACD. The lengths of three of the sides are AB=6 cmAB = 6\text{ cm}, BC=8 cmBC = 8\text{ cm}, and CD=12 cmCD = 12\text{ cm}. If the length of side ADAD is an integer k cmk\text{ cm}, what is the sum of the minimum and maximum possible values of kk?

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Answer: 26

Answer

26
To find the minimum and maximum values of the integer side kk, we first determine the range of the shared diagonal ACAC. In the bottom triangle, the Triangle Inequality Theorem requires 2<AC<142 < AC < 14. In the top triangle, the theorem requires ∣12−AC∣<k<12+AC|12 - AC| < k < 12 + AC. To minimize kk, we look at the lower bound ∣12−AC∣|12 - AC|. Since ACAC can be 1212, the lower bound can be 00, meaning k>0k > 0, so the minimum integer value is 11. To maximize kk, we look at the upper bound 12+AC12 + AC. Since AC<14AC < 14, the upper bound is k<26k < 26, so the maximum integer value is 2525. The sum of these values is 1+25=261 + 25 = 26.

Step-by-Step Solution

1
Find the possible range of lengths for the diagonal ACAC using △ABC\triangle ABC.
2 cm<AC<14 cm2\text{ cm} < AC < 14\text{ cm}
According to the Triangle Inequality Theorem, the length of side ACAC must be greater than ∣BC−AB∣=8−6=2 cm|BC - AB| = 8 - 6 = 2\text{ cm} and less than BC+AB=8+6=14 cmBC + AB = 8 + 6 = 14\text{ cm}.
2
Set up the inequality for side AD=kAD = k in △ACD\triangle ACD.
∣12−AC∣<k<12+AC|12 - AC| < k < 12 + AC
By the Triangle Inequality Theorem applied to △ACD\triangle ACD, the length kk must be greater than the absolute difference ∣CD−AC∣|CD - AC| and less than the sum CD+ACCD + AC.
3
Find the minimum possible integer value for kk.
k=1k = 1
Since ACAC can be any real number between 22 and 1414, we can choose AC=12AC = 12, which makes the lower bound ∣12−12∣=0|12 - 12| = 0. Thus, we have k>0k > 0. The smallest integer greater than 00 is 11. We verify that k=1k = 1 is possible by choosing AC=12AC = 12, which satisfies the inequalities for both triangles.
4
Find the maximum possible integer value for kk.
k=25k = 25
Since AC<14AC < 14, the upper bound is k<12+AC<12+14=26k < 12 + AC < 12 + 14 = 26. Thus, we have k<26k < 26. The largest integer less than 2626 is 2525. We verify that k=25k = 25 is possible by choosing AC=13.5AC = 13.5, which satisfies the inequalities for both triangles.
5
Calculate the sum of the minimum and maximum possible integer values of kk.
1+25=261 + 25 = 26
To find the sum of the minimum and maximum values of kk as requested by the question.

Key Concept

The Triangle Inequality Theorem states that for any triangle, the sum of the lengths of any two sides must be strictly greater than the length of the remaining side.
Question 9Question →

A triangle has two sides of length 77 centimeters and 1010 centimeters. If the length of the third side, in centimeters, must be an integer, what is the minimum possible length of the third side?

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Answer: 4

Answer

4 centimeters
The Triangle Inequality Theorem states that the length of the third side, xx, must be strictly greater than the difference of the two known sides (10−7=310 - 7 = 3) and strictly less than their sum (10+7=1710 + 7 = 17). Therefore, 3<x<173 < x < 17. The smallest integer value that satisfies this inequality is 4.

Step-by-Step Solution

1
Identify the given side lengths of the triangle.
The two given side lengths are 7 centimeters and 10 centimeters.
These are the values needed to apply the Triangle Inequality Theorem.
2
Apply the Triangle Inequality Theorem to find the range of possible lengths for the third side, xx.
10−7<x<10+710 - 7 < x < 10 + 7, which simplifies to 3<x<173 < x < 17.
The theorem states that the length of any side of a triangle must be strictly greater than the difference of the other two sides and strictly less than their sum.
3
Identify the minimum integer value within the valid range.
The smallest integer strictly greater than 3 is 4.
The question specifies that the third side length must be an integer and asks for the minimum possible value.

Key Concept

The Triangle Inequality Theorem states that for any triangle with sides aa, bb, and cc, the length of the third side must satisfy ∣a−b∣<c<a+b|a - b| < c < a + b.
Estimated Time:1m 0s
Question 10Question →

In △ABC\triangle ABC, point DD lies on side ACAC. If AB=BDAB = BD, the measure of ∠A\angle A is 70∘70^\circ, and the measure of ∠C\angle C is 25∘25^\circ, what is the measure, in degrees, of ∠DBC\angle DBC?

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Answer: 45

Answer

The measure of ∠DBC\angle DBC is 45 degrees.
In △ABD\triangle ABD, since AB=BDAB = BD, the triangle is isosceles and the base angles opposite these sides are equal. Therefore, the measure of ∠ADB\angle ADB is equal to the measure of ∠A\angle A, which is 70∘70^\circ. Because points AA, DD, and CC form a straight line, the angles ∠ADB\angle ADB and ∠BDC\angle BDC are supplementary, meaning the measure of ∠BDC=180∘−70∘=110∘\angle BDC = 180^\circ - 70^\circ = 110^\circ. The sum of the interior angles in △BCD\triangle BCD must be 180∘180^\circ, so the measure of ∠DBC=180∘−110∘−25∘=45∘\angle DBC = 180^\circ - 110^\circ - 25^\circ = 45^\circ.

Step-by-Step Solution

1
Determine the measure of ∠ADB\angle ADB using the properties of isosceles triangle △ABD\triangle ABD.
The measure of ∠ADB\angle ADB is 70∘70^\circ.
Since AB=BDAB = BD, the angles opposite those sides, ∠A\angle A and ∠ADB\angle ADB, are equal.
2
Calculate the measure of the supplementary angle ∠BDC\angle BDC.
The measure of ∠BDC\angle BDC is 110∘110^\circ.
Points AA, DD, and CC are collinear, meaning ∠ADB\angle ADB and ∠BDC\angle BDC form a linear pair and sum to 180∘180^\circ.
3
Determine the measure of ∠DBC\angle DBC using the triangle angle sum theorem on △BCD\triangle BCD.
The measure of ∠DBC\angle DBC is 45∘45^\circ.
The sum of the interior angles of △BCD\triangle BCD is 180∘180^\circ, so the measure of ∠DBC\angle DBC is 180∘−(110∘+25∘)=45∘180^\circ - (110^\circ + 25^\circ) = 45^\circ.

Key Concept

Using properties of isosceles triangles, linear pairs, and the triangle angle sum theorem to trace unknown angles.
Question 11Question →

An architect is designing a triangular window with side lengths, in feet, represented by xx, 3x−23x - 2, and 1818. If the value of xx must be an integer, what is the sum of all possible values of xx?

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Answer: 30

Answer

The sum of all possible integer values of xx is 30.
By applying the Triangle Inequality Theorem, the sum of any two sides of a triangle must be strictly greater than the third side. This yields the inequalities x+(3x−2)>18x + (3x - 2) > 18 (which simplifies to x>5x > 5), x+18>3x−2x + 18 > 3x - 2 (which simplifies to x<10x < 10), and (3x−2)+18>x(3x - 2) + 18 > x (which simplifies to x>−8x > -8). The intersection of these inequalities is 5<x<105 < x < 10. Since xx must be an integer, the possible values are 66, 77, 88, and 99. The sum of these values is 6+7+8+9=306 + 7 + 8 + 9 = 30.

Step-by-Step Solution

1
Set up the first triangle inequality constraint where the sum of the two variable sides is greater than the constant side.
x+(3x−2)>18  ⟹  4x>20  ⟹  x>5x + (3x - 2) > 18 \implies 4x > 20 \implies x > 5
The Triangle Inequality Theorem states that the sum of any two sides of a triangle must be strictly greater than the third side.
2
Set up the second triangle inequality constraint where the sum of xx and the constant side is greater than the other variable side.
x+18>3x−2  ⟹  20>2x  ⟹  x<10x + 18 > 3x - 2 \implies 20 > 2x \implies x < 10
To satisfy the Triangle Inequality Theorem for all combinations of sides.
3
Set up the third triangle inequality constraint where the sum of the second variable side and the constant side is greater than the first variable side.
(3x−2)+18>x  ⟹  2x>−16  ⟹  x>−8(3x - 2) + 18 > x \implies 2x > -16 \implies x > -8
To ensure the third side combination is mathematically valid.
4
Find the intersection of all three inequalities to determine the valid range for xx.
5<x<105 < x < 10
The value of xx must satisfy all three inequalities simultaneously.
5
Identify the integer values of xx within the open interval (5,10)(5, 10) and calculate their sum.
The integers are 6,7,8,6, 7, 8, and 99. Sum = 6+7+8+9=306 + 7 + 8 + 9 = 30.
The problem specifies that xx must be an integer, so we sum only the integers strictly between 5 and 10.

Key Concept

Triangle Inequality Theorem
Estimated Time:2m 0s
Question 12Question →

In △XYZ\triangle XYZ, the measure of ∠X\angle X is 40∘40^\circ. If the measure of ∠Y\angle Y is three times the measure of ∠X\angle X, what is the measure, in degrees, of ∠Z\angle Z?

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Answer: 20

Answer

The measure of ∠Z\angle Z is 2020 degrees.
To find the measure of ∠Z\angle Z, we first determine the measure of ∠Y\angle Y. Since ∠Y\angle Y is three times the measure of ∠X\angle X (40∘40^\circ), we calculate 3×40∘=120∘3 \times 40^\circ = 120^\circ. Because the interior angles of a triangle must sum to 180∘180^\circ, the remaining angle ∠Z\angle Z is found by subtracting the measures of ∠X\angle X and ∠Y\angle Y from 180∘180^\circ: 180∘−40∘−120∘=20∘180^\circ - 40^\circ - 120^\circ = 20^\circ.

Step-by-Step Solution

1
Calculate the measure of ∠Y\angle Y.
The measure of ∠Y\angle Y is 120∘120^\circ.
The measure of ∠Y\angle Y is specified to be three times the measure of ∠X\angle X, which is given as 40∘40^\circ. Multiplying 40∘40^\circ by 33 gives 120∘120^\circ.
2
Calculate the measure of ∠Z\angle Z.
The measure of ∠Z\angle Z is 20∘20^\circ.
The interior angles of any triangle sum to 180∘180^\circ. Subtracting the sum of the measures of ∠X\angle X (40∘40^\circ) and ∠Y\angle Y (120∘120^\circ) from 180∘180^\circ yields the measure of ∠Z\angle Z.

Key Concept

The sum of the interior angles of a triangle is always 180∘180^\circ.
Estimated Time:45s
Question 13Question →

The measures of the interior angles of a triangle are in the ratio 2:3:52:3:5. What is the measure, in degrees, of the largest exterior angle of this triangle?

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Answer: 144∘144^\circ

Answer

144∘144^\circ
The sum of the interior angles of a triangle is 180∘180^\circ. Given the ratio 2:3:52:3:5, the angles are 2x2x, 3x3x, and 5x5x. Adding these gives 10x=180∘10x = 180^\circ, which solves to x=18∘x = 18^\circ. The smallest interior angle is 2(18∘)=36∘2(18^\circ) = 36^\circ. Since an exterior angle is supplementary to its adjacent interior angle, the largest exterior angle is the supplement of the smallest interior angle: 180∘−36∘=144∘180^\circ - 36^\circ = 144^\circ.

Step-by-Step Solution

1
Represent the three interior angles of the triangle in terms of a variable xx based on the given ratio.
The angles can be expressed as 2x2x, 3x3x, and 5x5x.
A ratio of 2:3:52:3:5 means the angles are multiples of a common factor xx.
2
Set up and solve an equation for xx using the triangle angle sum theorem.
2x+3x+5x=180∘  ⟹  10x=180∘  ⟹  x=18∘2x + 3x + 5x = 180^\circ \implies 10x = 180^\circ \implies x = 18^\circ.
The sum of the interior angles of any triangle is always 180∘180^\circ.
3
Determine the measures of all three interior angles.
The angles are 2(18∘)=36∘2(18^\circ) = 36^\circ, 3(18∘)=54∘3(18^\circ) = 54^\circ, and 5(18∘)=90∘5(18^\circ) = 90^\circ.
This identifies the individual interior angle measures to find the smallest one.
4
Calculate the measure of the largest exterior angle by finding the supplement of the smallest interior angle.
180∘−36∘=144∘180^\circ - 36^\circ = 144^\circ.
An interior angle and its adjacent exterior angle form a linear pair and sum to 180∘180^\circ. To maximize the exterior angle, we must subtract the smallest interior angle.

Key Concept

Interior and exterior angle relationships in triangles and ratio partitioning

Alternative Method

By the Exterior Angle Theorem, an exterior angle of a triangle equals the sum of the two non-adjacent interior angles. To find the largest exterior angle, sum the two largest interior angles. The two largest interior angles in ratio units are 33 parts and 55 parts, summing to 88 parts. Since the total sum of the interior angles is 1010 parts (180∘180^\circ), each part is 18∘18^\circ. Thus, the largest exterior angle is 8×18∘=144∘8 \times 18^\circ = 144^\circ.
Estimated Time:1m 0s
Question 14Question →

In △ABC\triangle ABC, the measures of the three exterior angles (one at each vertex) are in the ratio 4:5:64:5:6. What is the measure of the smallest interior angle of △ABC\triangle ABC?

Show answer & explanation

Answer: 36∘36^\circ

Answer

The correct answer is 36∘36^\circ.
The correct answer is 36∘36^\circ. The sum of the three exterior angles of a triangle is 360∘360^\circ. Given the ratio 4:5:64:5:6, we set up the equation 4k+5k+6k=360∘4k + 5k + 6k = 360^\circ, which simplifies to 15k=360∘15k = 360^\circ, yielding k=24∘k = 24^\circ. The largest exterior angle is 6(24∘)=144∘6(24^\circ) = 144^\circ. Since the interior and exterior angles at a vertex are supplementary, the smallest interior angle corresponds to the largest exterior angle, which is 180∘−144∘=36∘180^\circ - 144^\circ = 36^\circ.

Step-by-Step Solution

1
Recall the sum of the exterior angles of a triangle.
The sum of the exterior angles of any convex polygon (including a triangle) is 360∘360^\circ.
This is a fundamental geometric property of exterior angles.
2
Set up an algebraic equation to find the value of each part in the ratio.
Let the measures of the exterior angles be 4k4k, 5k5k, and 6k6k. Then: 4k+5k+6k=360∘  ⟹  15k=360∘  ⟹  k=24∘4k + 5k + 6k = 360^\circ \implies 15k = 360^\circ \implies k = 24^\circ.
The sum of the ratio parts must equal the total sum of the exterior angles.
3
Identify which exterior angle corresponds to the smallest interior angle.
Since an interior angle and its adjacent exterior angle are supplementary (180∘180^\circ sum), the smallest interior angle must correspond to the largest exterior angle, which is 6k=6(24∘)=144∘6k = 6(24^\circ) = 144^\circ.
As the exterior angle increases, the supplementary interior angle decreases.
4
Calculate the smallest interior angle.
180∘−144∘=36∘180^\circ - 144^\circ = 36^\circ.
Subtracting the largest exterior angle from 180∘180^\circ yields the smallest interior angle.

Key Concept

The sum of the exterior angles of a triangle is 360∘360^\circ, and the interior and exterior angles at each vertex are supplementary.
Estimated Time:1m 30s
Question 15Question →

An exterior angle of a triangle measures 135∘135^\circ. The two nonadjacent interior angles of the triangle have measures in the ratio 2:32:3. What is the measure, in degrees, of the largest interior angle of the triangle?

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Answer: 81

Answer

The measure of the largest interior angle of the triangle is 81 degrees.
According to the Exterior Angle Theorem, the sum of the two nonadjacent interior angles equals the measure of the exterior angle: 2x+3x=1352x + 3x = 135, which simplifies to 5x=1355x = 135 and gives x=27x = 27. The nonadjacent interior angles are 2(27)=54∘2(27) = 54^\circ and 3(27)=81∘3(27) = 81^\circ. The adjacent interior angle is 180∘−135∘=45∘180^\circ - 135^\circ = 45^\circ. Comparing the three angles (45∘45^\circ, 54∘54^\circ, 81∘81^\circ), the largest is 81∘81^\circ.

Step-by-Step Solution

1
Use the Exterior Angle Theorem to relate the two nonadjacent interior angles to the given exterior angle.
Equation: 2x+3x=1352x + 3x = 135
The theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of its two nonadjacent interior angles.
2
Solve the linear equation for xx.
x=27x = 27
Combine like terms to get 5x=1355x = 135, then divide both sides by 5.
3
Calculate the measures of the two nonadjacent interior angles.
First angle: 2(27)=54∘2(27) = 54^\circ; Second angle: 3(27)=81∘3(27) = 81^\circ
Multiply each part of the ratio by the scale factor x=27x = 27.
4
Calculate the measure of the remaining interior angle.
Adjacent angle: 180∘−135∘=45∘180^\circ - 135^\circ = 45^\circ
An interior angle and its adjacent exterior angle lie on a straight line and are supplementary (they sum to 180∘180^\circ).
5
Compare the three interior angle measures to find the largest.
The largest angle is 81∘81^\circ.
Comparing 45∘45^\circ, 54∘54^\circ, and 81∘81^\circ shows that 81∘81^\circ is the maximum value.

Key Concept

Exterior Angle Theorem and interior angle relationships in a triangle

Alternative Method

Find the adjacent interior angle first: 180∘−135∘=45∘180^\circ - 135^\circ = 45^\circ. Since the sum of all interior angles in a triangle is 180∘180^\circ, the sum of the remaining two interior angles must be 180∘−45∘=135∘180^\circ - 45^\circ = 135^\circ. Set up the ratio equation 2x+3x=1352x + 3x = 135 and solve for xx to find the other two angles.
Estimated Time:1m 15s
Question 16Question →

In △ABC\triangle ABC, the measure of interior angle ∠A\angle A is 45∘45^\circ, and the exterior angle at vertex BB measures 125∘125^\circ. What is the measure, in degrees, of interior angle ∠C\angle C?

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Answer: 80

Answer

The measure of interior angle ∠C\angle C is 8080 degrees.
According to the Exterior Angle Theorem, the measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles. In this case, the exterior angle at vertex BB measures 125∘125^\circ, and one of its remote interior angles, ∠A\angle A, measures 45∘45^\circ. The other remote interior angle is ∠C\angle C. Setting up the equation: 125∘=45∘+m∠C125^\circ = 45^\circ + \text{m}\angle C. Solving for the measure of ∠C\angle C gives 125∘−45∘=80∘125^\circ - 45^\circ = 80^\circ.

Step-by-Step Solution

1
Use the Exterior Angle Theorem to relate the given angles.
The exterior angle at vertex BB (125∘125^\circ) is equal to the sum of the remote interior angles, ∠A\angle A and ∠C\angle C. This gives the equation: 125∘=45∘+m∠C125^\circ = 45^\circ + \text{m}\angle C.
The Exterior Angle Theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles.
2
Solve the equation for the measure of ∠C\angle C.
m∠C=125∘−45∘=80∘\text{m}\angle C = 125^\circ - 45^\circ = 80^\circ
Subtract 45∘45^\circ from both sides of the equation to isolate the measure of ∠C\angle C.

Key Concept

Exterior Angle Theorem
Question 17Question →

In acute triangle ABCABC, the measure of ∠A\angle A is 74∘74^\circ. The altitude from vertex BB to side ACAC and the altitude from vertex CC to side ABAB intersect at point HH inside the triangle. What is the measure, in degrees, of ∠BHC\angle BHC?

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Answer: 106

Answer

The measure of ∠BHC\angle BHC is 106 degrees.
In quadrilateral AEHDAEHD, the sum of the angles is 360∘360^\circ. Since the angles at EE and DD are 90∘90^\circ because they are formed by altitudes, the sum of ∠A\angle A and ∠EHD\angle EHD must be 180∘180^\circ. Thus, ∠EHD=180∘−74∘=106∘\angle EHD = 180^\circ - 74^\circ = 106^\circ. Since ∠BHC\angle BHC and ∠EHD\angle EHD are vertical angles, ∠BHC=106∘\angle BHC = 106^\circ.

Step-by-Step Solution

1
Define the intersection points of the altitudes with the opposite sides.
Let the altitude from vertex BB intersect side ACAC at point DD, and let the altitude from vertex CC intersect side ABAB at point EE. Therefore, ∠ADH=90∘\angle ADH = 90^\circ and ∠AEH=90∘\angle AEH = 90^\circ.
Altitudes by definition are perpendicular to the sides they intersect, forming 90∘90^\circ angles.
2
Analyze the sum of interior angles in the quadrilateral AEHDAEHD.
The sum of the interior angles of a quadrilateral is 360∘360^\circ, so ∠A+∠AEH+∠EHD+∠ADH=360∘\angle A + \angle AEH + \angle EHD + \angle ADH = 360^\circ.
Any quadrilateral can be split into two triangles, making the sum of its interior angles 360∘360^\circ.
3
Substitute the known angle measures to find the measure of ∠EHD\angle EHD.
74∘+90∘+∠EHD+90∘=360∘  ⟹  254∘+∠EHD=360∘  ⟹  ∠EHD=106∘74^\circ + 90^\circ + \angle EHD + 90^\circ = 360^\circ \implies 254^\circ + \angle EHD = 360^\circ \implies \angle EHD = 106^\circ.
Solving the linear equation for the unknown angle ∠EHD\angle EHD.
4
Relate ∠EHD\angle EHD to the target angle ∠BHC\angle BHC.
Since line segments BDBD and CECE intersect at HH, the angles ∠BHC\angle BHC and ∠EHD\angle EHD are vertical angles, so ∠BHC=∠EHD=106∘\angle BHC = \angle EHD = 106^\circ.
Vertical angles are equal in measure.

Key Concept

The sum of angles in quadrilaterals, the definition of altitudes, and vertical angles within triangles.
Estimated Time:2m 0s
Question 18Question →

A triangle has side lengths of 77, 1212, and 2x+12x + 1. If xx is an integer, how many possible values of xx exist?

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Answer: 66

Answer

6
To form a valid triangle, the length of any side must be strictly less than the sum of the other two sides and strictly greater than the positive difference of the other two sides. Applying this to the side lengths 77, 1212, and 2x+12x+1 gives the inequality 12−7<2x+1<12+712 - 7 < 2x + 1 < 12 + 7, which simplifies to 5<2x+1<195 < 2x + 1 < 19. Subtracting 11 from all parts gives 4<2x<184 < 2x < 18, and dividing by 22 gives 2<x<92 < x < 9. The integers in this open interval are 3,4,5,6,7,3, 4, 5, 6, 7, and 88, which counts to 6 possible values.

Step-by-Step Solution

1
Apply the Triangle Inequality Theorem, which states that the sum of any two sides of a triangle must be strictly greater than the third side.
We obtain three inequalities: (1) 7+12>2x+17 + 12 > 2x + 1, (2) 7+(2x+1)>127 + (2x + 1) > 12, and (3) 12+(2x+1)>712 + (2x + 1) > 7.
To find the valid range for the unknown side length expression 2x+12x + 1.
2
Solve the three inequalities for xx.
From (1), 18>2x  ⟹  x<918 > 2x \implies x < 9. From (2), 2x+8>12  ⟹  2x>4  ⟹  x>22x + 8 > 12 \implies 2x > 4 \implies x > 2. From (3), 2x+13>7  ⟹  2x>−6  ⟹  x>−32x + 13 > 7 \implies 2x > -6 \implies x > -3. Combining the most restrictive bounds gives the interval 2<x<92 < x < 9.
To isolate the variable xx and establish its upper and lower bounds.
3
Identify and count all integers xx that satisfy the inequality 2<x<92 < x < 9.
The integers strictly between 2 and 9 are 3,4,5,6,73, 4, 5, 6, 7, and 88. There are 6 such integers.
To find the number of possible integer values for xx as requested by the question.

Key Concept

Triangle Inequality Theorem
Question 19Question →

A triangle has side lengths such that one side is 33 units less than twice an integer yy, another side is 44 units more than yy, and the third side is 1111 units. How many different triangles can be formed with these side lengths?

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Answer: 14

Answer

14
The correct answer is 14. Applying the Triangle Inequality Theorem, the sum of any two sides of the triangle must be strictly greater than the third side. This gives three inequalities: (2y - 3) + (y + 4) > 11, (2y - 3) + 11 > y + 4, and (y + 4) + 11 > 2y - 3. Solving these inequalities yields y > 10/3, y > -4, and y < 18. The common intersection is 10/3 < y < 18. Since y is an integer, y can range from 4 to 17 inclusive. There are 17 - 4 + 1 = 14 integers in this range.

Step-by-Step Solution

1
Translate the verbal descriptions into algebraic expressions for the side lengths.
The three side lengths are represented as 2y−32y - 3, y+4y + 4, and 1111.
To apply mathematical theorems, the side lengths must first be written in algebraic form.
2
Set up the three inequalities required by the Triangle Inequality Theorem, stating that the sum of any two sides must be strictly greater than the third side.
The inequalities are: 1) (2y−3)+(y+4)>11(2y - 3) + (y + 4) > 11, 2) (2y−3)+11>y+4(2y - 3) + 11 > y + 4, and 3) (y+4)+11>2y−3(y + 4) + 11 > 2y - 3.
The Triangle Inequality Theorem guarantees that three segment lengths can form a non-degenerate triangle.
3
Solve each of the three inequalities for yy.
1) 3y+1>11  ⟹  3y>10  ⟹  y>103≈3.333y + 1 > 11 \implies 3y > 10 \implies y > \frac{10}{3} \approx 3.33.
2) 2y+8>y+4  ⟹  y>−42y + 8 > y + 4 \implies y > -4.
3) y+15>2y−3  ⟹  18>y  ⟹  y<18y + 15 > 2y - 3 \implies 18 > y \implies y < 18.
Solving the inequalities identifies the constraints on the variable yy.
4
Find the intersection of all three solution intervals and identify the valid integer values for yy.
The intersection of the intervals is 3.33<y<183.33 < y < 18. Since yy must be an integer, yy can be any integer from 44 to 1717, inclusive: {4,5,6,7,8,9,10,11,12,13,14,15,16,17}\{4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17\}.
The value of yy must simultaneously satisfy all three inequality conditions.
5
Count the number of integers in the inclusive range [4,17][4, 17].
The number of integers is 17−4+1=1417 - 4 + 1 = 14.
This gives the total number of distinct triangles that can be formed.

Key Concept

Triangle Inequality Theorem
Question 20Question →

In △PQR\triangle PQR, the measure of interior angle ∠P\angle P is 40∘40^\circ. If the measure of ∠Q\angle Q is twice the measure of ∠P\angle P, what is the measure, in degrees, of the interior angle ∠R\angle R?

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Answer: 60

Answer

The measure of interior angle ∠R\angle R is 60 degrees.
By definition, the interior angles of any triangle sum to 180 degrees. Given that angle P is 40 degrees, and angle Q is twice angle P (80 degrees), the sum of angles P and Q is 120 degrees. Subtracting this from 180 degrees gives a remaining measure of 60 degrees for angle R.

Step-by-Step Solution

1
Calculate the measure of angle Q
m∠Q=80∘m\angle Q = 80^\circ
The problem states that the measure of angle Q is twice the measure of angle P, which is 40 degrees.
2
Apply the Triangle Angle Sum Theorem
m\angle P + m\angle Q + m\angle R = 180^\circ
The sum of the measures of the interior angles of any triangle is always 180 degrees.
3
Solve for the measure of angle R
m∠R=60∘m\angle R = 60^\circ
Substitute the values of angles P and Q into the equation: 40 + 80 + m\angle R = 180, which simplifies to 120 + m\angle R = 180, and solving for m\angle R gives 60.

Key Concept

The sum of the measures of the interior angles of a triangle is always 180 degrees.
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Triangle Properties and Angle Theorems Practice Questions — ACT | Examkin