Question

vasquez unit 5

1. what additional congruence must be established to prove \\( \triangle c a t \cong \triangle s a t \\) by hl?
2. what additional congruence must be established to prove \\( \triangle t i e \cong \triangle m e i \\) by aas?

Explanation:

3.

Step1: Recall HL (Hypotenuse - Leg) theorem

HL theorem states that for right - angled triangles, if the hypotenuse and one leg of a right - triangle are congruent to the hypotenuse and one leg of another right - triangle, then the two right - triangles are congruent. In \(\triangle CAT\) and \(\triangle SAT\), \(\angle CTA=\angle STA = 90^{\circ}\) (right angles) and \(AT\) is a common leg (\(AT\cong AT\)).

Step2: Identify the hypotenuse

The hypotenuses of \(\triangle CAT\) and \(\triangle SAT\) are \(CA\) and \(SA\) respectively. So, we need \(CA\cong SA\) to prove \(\triangle CAT\cong\triangle SAT\) by HL.

4.

Step1: Recall AAS (Angle - Angle - Side) theorem

AAS theorem states that if two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle, then the two triangles are congruent. In \(\triangle TIE\) and \(\triangle MEI\), \(\angle TIE=\angle MEI\) (given by the mark) and \(\angle IET=\angle IEM\) (given by the mark).

Step2: Identify the non - included side

The non - included sides for the pairs of angles are \(TE\) and \(MI\). So, we need \(TE\cong MI\) to prove \(\triangle TIE\cong\triangle MEI\) by AAS.

Answer:

1. \(CA\cong SA\)
2. \(TE\cong MI\)