a What parts are corresponding in the two quadrilaterals?
B
b What pair of angles do ∠ GBE and ∠ GDE form with ∠ ABE and ∠ CDE?
C
c What pair of angles do ∠ GBE form with ∠ GED?
D
d What's unique about one of the triangle's sides?
A
a See solution
B
b See solution
C
c See solution
D
d See solution
Practice makes perfect
a Let's highlight the two quadrilaterals that we know are congruent, ABEF and CDEF.
When two figures are congruent, corresponding parts are congruent. Let's mark BE and DE as well as ∠ ABE and ∠ CDE in our diagram.
b From part A, we concluded that ∠ ABE≅ ∠ CDE. If we look at the diagram again, we see that ∠ GBE forms a linear pair with ∠ ABE. Similarly, ∠ GDE forms a linear pair with ∠ CDE. According to the Linear Pair Postulate, we know they are supplementary angles which means we can write the following equations:
&m∠ ABE+m∠ GBE=180^(∘)
&m∠ CDE+m∠ GDE=180^(∘)
Since both angle measures equate to 180^(∘), we can use the Transitive Property of Equality to equate the left-hand sides and then show that the angle measures of ∠ GBE and ∠ GDF are the same.
Having proved that the angle measures of ∠ GBE and ∠ GDE are the same, we can, by the definition of congruent angles, say the following.
∠ GBE≅ ∠ GDE
Let's illustrate this in the diagram.
c From the diagram, we see that ∠ GED is a right angle. We also see that ∠ GED and ∠ GEB are a linear pair which means these angles are supplementary according to the Linear Pair Postulate. We can now write the following equation.
m∠ GEB+m∠ GED=180^(∘)
Since we know the measure of m∠ GED we can find the measure of m∠ GEB as well.
Since both angles are right angles, they are congruent.
d From previous parts we know that BE≅ DE. We also know that ∠ GEB≅ ∠ GED and ∠ GBE≅ ∠ GDE. Let's complete our diagram with this information and highlight the triangles we are working with.
To prove that these triangles are congruent, △ BEG ≅ △ DEG, we have to show that corresponding sides are congruent and that corresponding angles are congruent.
Sides
We already know that two sets of sides are congruent. The third side is shared by the triangles which means it has to be congruent as well by the Reflexive Property of Congruence. Therefore, we have three pairs of congruent sides.
Angles
As with the sides, we know that two of the angles are congruent. According to the Third Angles Theorem, if two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent. Therefore, we can say the third angle is congruent as well.
Therefore, we can say that we have enough information to prove that △ BEG ≅ △ DEG.
