Exercise 20 Page 257

Practice makes perfect

a We know that △ ABD and △ CBD are congruent equilateral triangles which means their sides all have the same length. Let's illustrate this in a diagram. We will also add a couple of angle measures that will come in handy in our solution.

How can we be sure that △ ABC is isosceles? Well, let's start by isolating the triangle we want to analyze.

As we can see from the diagram, the triangle has two congruent sides AB≅ CB, which fits the definition of an isosceles triangle.

b From part A, we know that △ ABC is isosceles. This means that the base angles are congruent, which just so happens to be the same angles as ∠ BAE and ∠ BCE.

c Let's isolate the two relevant triangles in the diagram.

Notice that BE is shared between the two triangles. Therefore, by the Reflexive Property of Congruence, we can say that this side is congruent in the two triangles.

By using the SAS Congruence Theorem, we can claim that △ ABE and △ CBE are congruent.

d Let's look at the diagram from part B.

The Triangle Sum Theorem states that the sum of the interior angles in a triangle is 180^(∘). Since this is an isosceles triangle, the base angles are congruent. Therefore, we can write the following equation 120^(∘)+m∠ BAE+m∠ BCE=180^(∘). Since m∠ BCE=m∠ BAE, we can use the Substitution Property of Equality and replace m∠ BCE with m∠ BAE and then solve for this angle.

120^(∘)+m∠ BAE+m∠ BCE=180^(∘)

Substitute

m∠ BAE= m∠ BCE

120^(∘)+m∠ BAE+ m∠ BAE=180^(∘)

▼

Solve for m∠ BAE

AddTerms

Add terms

120^(∘)+2m∠ BAE=180^(∘)

SubEqn

LHS-120^(∘)=RHS-120^(∘)

2m∠ BAE=60^(∘)