Question

In the figure below, ∠DEC ≅ ∠DCE, ∠B ≅ ∠F, and segment DF is congruent to segment BD. Point C is the point of intersection between segment AG and segment BD, while point E is the point of intersection between segment AG and segment DF.

Answer 1

Prove ΔABC ≅ ΔGFE.

Answer 2

I am very confused and cant figure it out

Answer 3

Well since <DEC=<DCE what type of triangle is DCE.

Answer 4

It is a isosceles

Answer 5

Good. So DC=DE. What two line segments make up DB? What two line segments make up DF?

Answer 6

DB= DC+CB and DF= DE+EF

Answer 7

Now knowing this can you prove CB=EF by substituting and subtracting?

Answer 8

Yes

Answer 9

Okay so we currently have one side and one angle. We can get the last angle for ASA congruence by using vertical angles. Can you see vertical angles that would help you prove that two of the angles of the 2 triangles are equal?

Answer 10

Angles B and F

Answer 11

Those aren't vertical angles.

Answer 12

I cant seem to find any vertical angles

Answer 13

Hint: One of pairs of vertical angles are <ACB and <DCE.

Answer 14

Can you find the other pair now?

Answer 15

Angles C and E?

Answer 16

Remember vertical angles are opposite angles formed by intersecting lines.

Answer 17

<GEF and <DCE

Answer 18

Good and what properties do vertical angles have?

Answer 19

Vertical angles are always congruent

Answer 20

Yep. And we're already given that <DCE=<DEC. So <ACB=<GEF and we have triangle ABC is congruent to triangle GFE.


BTW above did you mean <GEF and <DEC?

Answer 21

Yes sorry typed it in the wrong order

Answer 22

Okay so ABC is congruent to GFE by ASA.

Answer 23

Ok thank you so much