Question

Triangles abd and cbd are shown. Triangle abc is divided into two smaller triangles which are triangle abd and triangle dbc. Triangle abd and triangle dbc share a common side bd. Point d lies on segment ac. Segment ad is congruent to segment cd. If m∠bdc = 70°, what is the relationship between ab and bc?

1) ab = bc
2) ab < bc
3) ab > bc
4) ab bc < ac

Answer 1

In triangle ABC, the relationship between AB and BC is that they are congruent, which means they have the same length. Therefore, the correct answer is AB = BC.

Step-by-step explanation:

Since triangles ABD and BDC share a common side BD, they are adjoining triangles. We know that segment AD is congruent to segment CD, and the angle of triangle BDC is given as 70°. If we let the measure of angle ABD be x, then the measure of angle ADC will also be x, because AD is congruent to CD. We can use the fact that the sum of angles in a triangle is 180° to find the measure of angle ABC. The sum of angles ABC, ABD, and BDC is 180°. Substituting the values we know, we get:

x + x + 70 = 180

2x + 70 = 180

2x = 110

x = 55

Therefore, the measure of angle ABC is 180 - (55 + 70) = 180 - 125 = 55°.

We can conclude that angle ABC is equal to angle BDC, which means that triangle ABC is an isosceles triangle. In an isosceles triangle, the two legs are congruent. Therefore, we can say that AB is congruent to BC.

So the correct relationship between AB and BC is: 1) ab = bc