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Difficulty: MediumTriangle Congruence, Similarity, and Theorems

In triangle ABCABC, point DD lies on side BCBC. Line segment ADAD is drawn such that AB=ADAB = AD and AD=CDAD = CD. If the measure of angle BACBAC is 75∘75^\circ, what is the measure, in degrees, of angle BB?

Answer: 70 degrees

Answer

The measure of angle BB is 70∘70^\circ.
By representing the angles using the properties of the isosceles triangles ABDABD and ADCADC, we set up a system of equations where ∠B=x\angle B = x and ∠C=y\angle C = y such that x=2yx = 2y. Using the angle sum theorem and angle addition, we find that (180∘−2x)+y=75∘(180^\circ - 2x) + y = 75^\circ. Substituting x=2yx = 2y yields y=35∘y = 35^\circ, and thus ∠B=x=70∘\angle B = x = 70^\circ.

Step-by-Step Solution

1
Set up base angles for the isosceles triangle ABDABD.
Let ∠B=∠ADB=x\angle B = \angle ADB = x.
Because AB=ADAB = AD, triangle ABDABD is an isosceles triangle, making its base angles equal.
2
Express the measure of angle ADCADC in terms of xx.
∠ADC=180∘−x\angle ADC = 180^\circ - x.
Angles ∠ADB\angle ADB and ∠ADC\angle ADC form a linear pair along the line segment BCBC.
3
Establish the relationship between xx and yy using triangle ADCADC.
x=2yx = 2y, where y=∠DAC=∠Cy = \angle DAC = \angle C.
Since AD=CDAD = CD, triangle ADCADC is isosceles. The sum of angles in △ADC\triangle ADC is (180∘−x)+y+y=180∘(180^\circ - x) + y + y = 180^\circ, which simplifies to x=2yx = 2y.
4
Write the equation for the total measure of angle BACBAC.
(180∘−2x)+y=75∘(180^\circ - 2x) + y = 75^\circ.
The angle addition postulate states that ∠BAC=∠BAD+∠DAC\angle BAC = \angle BAD + \angle DAC. Since ∠BAD=180∘−2x\angle BAD = 180^\circ - 2x (from the sum of angles in △ABD\triangle ABD) and ∠DAC=y\angle DAC = y, this equals 75∘75^\circ.
5
Solve the system of equations for yy.
y=35∘y = 35^\circ.
Substituting x=2yx = 2y into (180∘−2x)+y=75∘(180^\circ - 2x) + y = 75^\circ yields 180∘−3y=75∘180^\circ - 3y = 75^\circ, which gives 3y=105∘3y = 105^\circ, so y=35∘y = 35^\circ.
6
Find the measure of angle BB.
∠B=70∘\angle B = 70^\circ.
Since ∠B=x\angle B = x and x=2yx = 2y, we have x=2(35∘)=70∘x = 2(35^\circ) = 70^\circ.

Key Concept

Using properties of isosceles triangles and angle sum theorems to solve for angle measures.
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Topics

Triangle Congruence, Similarity, and Theorems

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