Question

given: \\( \overline{ea} \cong \overline{eu} \\), \\( \overline{au} \parallel \overline{io} \\)
prove: \\( \triangle ieo \\) isosceles.

1. click here to insert
2. given
3. click here to insert
4. base angles theorem
5. \\( \overline{au} \parallel \overline{io} \\)
6. click here to insert

Explanation:

Step1: State the given

$\overline{EA}\cong\overline{EU}$

Step2: Use the Base Angles Theorem

$\angle1\cong\angle3$ (Base Angles Theorem states that in an isosceles triangle, the base angles are congruent. Since $\triangle EAU$ has $\overline{EA}\cong\overline{EU}$, $\angle1$ and $\angle3$ are base angles)

Step3: Use the property of parallel lines

$\angle1\cong\angle2$ (Alternate - interior angles, because $\overline{AU}\parallel\overline{IO}$ and $\overline{EI}$ is a transversal)
$\angle3\cong\angle4$ (Alternate - interior angles, because $\overline{AU}\parallel\overline{IO}$ and $\overline{EO}$ is a transversal)

Step4: Use the transitive property

Since $\angle1\cong\angle3$, $\angle1\cong\angle2$, and $\angle3\cong\angle4$, then $\angle2\cong\angle4$

Step5: Use the converse of the Base Angles Theorem

In $\triangle IEO$, if $\angle2\cong\angle4$, then $\overline{EI}\cong\overline{EO}$ (Converse of the Base Angles Theorem: If two angles of a triangle are congruent, then the sides opposite those angles are congruent). So, $\triangle IEO$ is isosceles.

Answer:

To prove $\triangle IEO$ is isosceles:

1. Given $\overline{EA}\cong\overline{EU}$
2. $\angle1\cong\angle3$ (Base Angles Theorem)
3. $\overline{AU}\parallel\overline{IO}$ (Given)

• $\angle1\cong\angle2$ (Alternate - interior angles)
• $\angle3\cong\angle4$ (Alternate - interior angles)

1. $\angle2\cong\angle4$ (Transitive property)
2. $\overline{EI}\cong\overline{EO}$ (Converse of the Base Angles Theorem), so $\triangle IEO$ is isosceles.