Question

1. \\( \overline { a b } \cong \overline { a c } \\) and \\( \overline { a y } \cong \overline { a x } \\) 2. definition of isosceles triangle
2. \\( a b = a c \\) and \\( a y = a x \\) 3. definition of congruency
3. \\( a y \cdot a c = a x \cdot a c \\) 4. multiplication property of equality
4. \\( a y \cdot a c = a x \cdot a b \\) 5. substitution property of equality
5. \\( \frac { a y \cdot a c } { a b } = a x \\) 6. division property of equality
6. \\( \frac { a y } { a b } = \frac { a x } { a c } \\) 7. division property of equality
7. ? 8. ?
8. \\( \triangle a b c \sim \triangle a y x \\) 9. sas similarity theorem

which statement and reason are missing in the proof?

Explanation:

Step1: Analyze the given information

We have \(AB = AC\) (from step 3), \(AY = AX\) (from step 3), and we are working towards proving \(\triangle ABC\sim\triangle AYX\) using the SAS (Side - Angle - Side) similarity theorem. The SAS similarity theorem states that if two sides of one triangle are proportional to two sides of another triangle and the included angles are equal, then the two triangles are similar.

Step2: Identify the included angle

In \(\triangle ABC\) and \(\triangle AYX\), \(\angle A\) is common to both triangles. So, \(\angle A=\angle A\) by the reflexive property of equality.

Answer:

1. \(\angle A=\angle A\); 8. reflexive property of equality