Question

lesson 23: base angles of isosceles triangles
exit ticket
for each of the following, if the given congruence exists, name the isosceles triangle and the pair of congruent angles for the triangle based on the image above.

1. $overline{ae} \cong \overline{le}$
2. $overline{le} \cong \overline{lg}$
3. $overline{an} \cong \overline{ln}$
4. $overline{en} \cong \overline{gn}$
5. $overline{ng} \cong \overline{lg}$
6. $overline{ae} \cong \overline{ne}$

Explanation:

Problem 1: $\boldsymbol{\overline{AE} \cong \overline{LE}}$

Step1: Identify the triangle

Since $\overline{AE} \cong \overline{LE}$, the triangle with these two sides as legs is $\triangle AEL$. In an isosceles triangle, the angles opposite the congruent sides are congruent.

Step2: Find congruent angles

The sides $\overline{AE}$ and $\overline{LE}$ are congruent, so the angles opposite them are $\angle ALE$ and $\angle LAE$ (or $\angle EAL$). So $\angle ALE \cong \angle LAE$.

Step1: Identify the triangle

Given $\overline{LE} \cong \overline{LG}$, the triangle is $\triangle LEG$.

Step2: Find congruent angles

The angles opposite $\overline{LE}$ and $\overline{LG}$ are $\angle LGE$ and $\angle LEG$. So $\angle LGE \cong \angle LEG$.

Step1: Identify the triangle

With $\overline{AN} \cong \overline{LN}$, the triangle is $\triangle ANL$.

Step2: Find congruent angles

The angles opposite $\overline{AN}$ and $\overline{LN}$ are $\angle ALN$ and $\angle NAL$. So $\angle ALN \cong \angle NAL$ (or $\angle LAN$).

Answer:

Isosceles triangle: $\triangle AEL$; Congruent angles: $\angle ALE \cong \angle LAE$ (or $\angle EAL$)

Problem 2: $\boldsymbol{\overline{LE} \cong \overline{LG}}$