Triangle Properties and Angle Theorems Practice Questions — ACT — Page 3

Question 41Question →

A triangle has side lengths of $x$ , $2x$ , and $15$ , where $x$ is an integer. What is the total number of possible values for $x$ ?

7

8

9

10

11

Show answer & explanation

Answer: 9

Answer

9

Applying the Triangle Inequality Theorem yields three inequalities:

x + 2x > 15

(which simplifies to

x > 5

),

x + 15 > 2x

(which simplifies to

x < 15

), and

2x + 15 > x

(which is always true since

x

is positive). Combining these results in

5 < x < 15

. The integers in this range are

\{6, 7, 8, 9, 10, 11, 12, 13, 14\}

. Counting them gives 9 possible values.

Step-by-Step Solution

1

Set up the three inequalities using the Triangle Inequality Theorem for the sides

x

,

2x

, and

15

.

The inequalities are

x + 2x > 15

,

x + 15 > 2x

, and

2x + 15 > x

.

The sum of the lengths of any two sides of a triangle must be strictly greater than the length of the third side.

2

Solve each inequality for

x

.

3x > 15 \implies x > 5

, and

15 > x \implies x < 15

. The third inequality

x > -15

is always true for positive side lengths.

To find the range of valid values for

x

.

3

Combine the inequalities and count the possible integer values for

x

.

The combined range is

5 < x < 15

. The integer values are

\{6, 7, 8, 9, 10, 11, 12, 13, 14\}

, which gives

14 - 6 + 1 = 9

values.

Since

x

is specified as an integer, we must identify and count all integers strictly between 5 and 15.

Key Concept

Triangle Inequality Theorem

Alternative Method

Instead of algebraic manipulation, we can test values of

x

directly. For

x=5

, the sides are

5, 10, 15

, but

5+10=15

, which does not form a triangle. For

x=15

, the sides are

15, 30, 15

, but

15+15=30

, which also does not form a triangle. Testing integer values between

5

and

15

confirms they all satisfy the triangle inequality, resulting in 9 valid integers.

Estimated Time:1m 0s

Question 42Question →

In the figure, line $L_1$ is parallel to line $L_2$ . Vertex $A$ of $\triangle ABC$ lies on $L_1$ , and vertices $B$ and $C$ lie on $L_2$ . Side $AB$ is perpendicular to $L_2$ . Point $D$ lies on $L_2$ such that $C$ is between $B$ and $D$ . If the measure of the exterior angle $\angle ACD$ is $132^\circ$ , what is the measure, in degrees, of $\angle BAC$ ?

Show answer & explanation

Answer: 42

Answer

42

The correct answer is

42

. Because

AB

is perpendicular to

L_2

,

\angle ABC

is

90^\circ

. The exterior angle

\angle ACD

is given as

132^\circ

, which means the adjacent interior angle

\angle ACB

must be supplementary to it:

180^\circ - 132^\circ = 48^\circ

. Since the interior angles of a triangle always sum to

180^\circ

, the remaining angle

\angle BAC

is

180^\circ - (90^\circ + 48^\circ) = 42^\circ

. Alternatively, applying the Exterior Angle Theorem, the exterior angle is equal to the sum of the two remote interior angles:

\angle ACD = \angle ABC + \angle BAC

, so

132^\circ = 90^\circ + \angle BAC

, which simplifies to

\angle BAC = 42^\circ

.

Step-by-Step Solution

1

Determine the measure of interior angle

\angle ABC

.

\angle ABC = 90^\circ

Since side

AB

is perpendicular to

L_2

, the angle it makes with

L_2

at vertex

B

is

90^\circ

.

2

Find the measure of interior angle

\angle ACB

.

\angle ACB = 48^\circ

The interior angle

\angle ACB

and the exterior angle

\angle ACD

form a linear pair along line

L_2

, making them supplementary:

\angle ACB = 180^\circ - 132^\circ = 48^\circ

.

3

Calculate the measure of

\angle BAC

using the angle sum of a triangle.

42^\circ

The interior angles of

\triangle ABC

sum to

180^\circ

. Subtracting the known angles gives

\angle BAC = 180^\circ - (90^\circ + 48^\circ) = 42^\circ

.

Key Concept

Triangle Angle Sum Theorem and Supplementary Angle Relationships

Question 43Question →

In $\triangle ABC$ , the lengths of the sides are $AB = 8$ , $BC = 11$ , and $AC = 14$ . Which of the following inequalities correctly compares the measures of the interior angles of $\triangle ABC$ ?

m\angle A < m\angle B < m\angle C

m\angle B < m\angle A < m\angle C

m\angle A < m\angle C < m\angle B

m\angle C < m\angle A < m\angle B

m\angle B < m\angle C < m\angle A

Show answer & explanation

Answer: $m\angle C < m\angle A < m\angle B$

Answer

The correct inequality is

m\angle C < m\angle A < m\angle B

.

In any triangle, the order of the measures of the interior angles matches the order of the lengths of their opposite sides. Since the side lengths are ordered

AB < BC < AC

(

8 < 11 < 14

), their opposite angles must be ordered in the same way. The angle opposite side

AB

is

\angle C

, the angle opposite side

BC

is

\angle A

, and the angle opposite side

AC

is

\angle B

. Therefore, the correct relationship is

m\angle C < m\angle A < m\angle B

.

Step-by-Step Solution

1

Identify the theorem relating triangle side lengths to their opposite angles.

The Side-Angle Relationship Theorem states that in any triangle, the order of the measures of the angles is the same as the order of the lengths of the sides opposite to those angles.

This establishes the rule needed to compare the angle measures based on the given side lengths.

2

Determine the opposite angle for each side length in

\triangle ABC

.

The angle opposite side

AB

is

\angle C

. The angle opposite side

BC

is

\angle A

. The angle opposite side

AC

is

\angle B

.

This maps each side length to the correct angle it controls.

3

Order the side lengths and apply the mapping to order the angle measures.

Since

8 < 11 < 14

, we write the side inequality as

AB < BC < AC

. Substituting the corresponding opposite angles gives

m\angle C < m\angle A < m\angle B

.

This yields the final correct inequality comparing the angle measures.

Key Concept

Triangle Side-Angle Relationship Theorem

Alternative Method

Another way to solve this is to sketch the triangle to scale. By drawing the longest side

AC = 14

as the horizontal base, the shortest side

AB = 8

on the left, and the side

BC = 11

on the right, it becomes visually apparent that the angle opposite the longest side (

\angle B

at the top vertex) is the largest angle, and the angle opposite the shortest side (

\angle C

at the bottom-right vertex) is the smallest angle.

Estimated Time:1m 0s

Question 44Question →

In $\triangle ABC$ , the measure of $\angle A$ is $40^\circ$ . The measure of $\angle B$ is three times the measure of $\angle C$ . What is the measure, in degrees, of $\angle B$ ?

Show answer & explanation

Answer: 105

Answer

The measure of angle B is 105 degrees.

The sum of the interior angles in any triangle is

180^\circ

. By setting the measure of

\angle C

to

x

and the measure of

\angle B

to

3x

, we can write the equation

40 + 3x + x = 180

. Solving for

x

yields

4x = 140

, which simplifies to

x = 35

. The measure of

\angle B

is

3x

, which is

3 \times 35 = 105^\circ

.

Step-by-Step Solution

1

Set up the equation using the fact that the sum of angles in a triangle is 180 degrees.

m\angle A + m\angle B + m\angle C = 180^\circ

The angles of any triangle in plane geometry sum to 180 degrees.

2

Represent the unknown angles algebraically.

Let

m\angle C = x

, then

m\angle B = 3x

. Substitute

m\angle A = 40^\circ

.

This allows solving for the unknown angles with a single-variable equation.

3

Solve the equation for

x

.

40 + 4x = 180 \implies 4x = 140 \implies x = 35

To find the measure of angle C.

4

Calculate the measure of angle B.

m\angle B = 3(35) = 105^\circ

Angle B is three times angle C, and we need to find the measure of angle B.

Key Concept

The sum of the interior angles of a triangle is always 180 degrees.

Question 45Question →

The measures of the three interior angles of a triangle are in the ratio $2:3:4$ . What is the measure, in degrees, of the smallest angle of the triangle?

20

40

60

72

80

Show answer & explanation

Answer: 40

Answer

40

The interior angles of any triangle sum to

180^\circ

. Given the ratio of the angles is

2:3:4

, the total number of ratio parts is

2 + 3 + 4 = 9

. Dividing the total degree sum by the number of parts gives

\frac{180^\circ}{9} = 20^\circ

per part. Since the smallest angle corresponds to the smallest part of the ratio, we multiply

2

by

20^\circ

to get

40^\circ

.

Step-by-Step Solution

1

Find the total number of parts in the ratio.

The total number of parts is

2 + 3 + 4 = 9

.

This determines how the

180^\circ

total sum of a triangle's interior angles is partitioned.

2

Calculate the value of a single part of the ratio.

One part is equal to

\frac{180^\circ}{9} = 20^\circ

.

The sum of the interior angles of any triangle is always

180^\circ

.

3

Find the measure of the smallest angle by multiplying by the smallest part of the ratio.

The smallest angle measures

2 \times 20^\circ = 40^\circ

.

The smallest angle corresponds to the smallest number in the ratio, which is 2.

Key Concept

Ratio-based angle partitioning in triangles

Question 46Question →

In $\triangle ABC$ , the measure of angle $A$ is $40^\circ$ . Point $D$ lies on side $AC$ such that segment $BD$ bisects angle $ABC$ . If the measure of angle $BDC$ is $75^\circ$ , what is the measure, in degrees, of angle $C$ ?

Show answer & explanation

Answer: 70

Answer

The measure of angle C is 70 degrees.

The correct measure of angle

C

is found by first identifying that angle

ADB

is supplementary to angle

BDC

, giving a measure of

105^\circ

. Using the triangle angle sum theorem on triangle

ABD

, we find that angle

ABD

is

35^\circ

. Since

BD

bisects angle

ABC

, angle

DBC

is also

35^\circ

. Finally, applying the triangle angle sum theorem to triangle

BCD

, we subtract the measures of angles

DBC

(

35^\circ

) and

BDC

(

75^\circ

) from

180^\circ

to get

70^\circ

.

Step-by-Step Solution

1

Find the measure of angle

ADB

using the supplementary angle relationship with angle

BDC

.

105^\circ

Angles

ADB

and

BDC

form a linear pair along the line segment

AC

, so their sum is

180^\circ

.

2

Find the measure of angle

ABD

using the sum of interior angles in

\triangle ABD

.

35^\circ

The sum of interior angles in any triangle is

180^\circ

. Therefore, the measure of angle

ABD

is

180^\circ - (40^\circ + 105^\circ) = 35^\circ

.

3

Find the measure of angle

DBC

using the definition of an angle bisector.

35^\circ

Since segment

BD

bisects angle

ABC

, the measures of angles

ABD

and

DBC

must be equal.

4

Find the measure of angle

C

using the sum of interior angles in

\triangle BCD

.

70^\circ

The sum of interior angles in

\triangle BCD

is

180^\circ

. Therefore, the measure of angle

C

is

180^\circ - (35^\circ + 75^\circ) = 70^\circ

.

Key Concept

Using the triangle angle sum theorem and angle bisector properties to determine unknown angle measures in a geometric figure.

Triangle Properties and Angle Theorems

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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Step-by-Step Solution

Key Concept

Common Mistakes

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Key Concept

Common Mistakes

Alternative Method

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Common Mistakes

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Common Mistakes

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Key Concept

Common Mistakes