In \triangle ABC, the measure of angle A is 40^\circ. Point D lies on side AC such that se… — ACT

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 $C$ ?

Answer: 70 degrees

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

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

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

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

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

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.

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

180^\circ

. Therefore, the measure of angle

C

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.

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic