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Difficulty: MediumTriangle Properties and Angle Theorems

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?

Answer: 45 degrees

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.
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Topics

Triangle Properties and Angle Theorems

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