Question

1. which of the following statements are invalid? choose all that apply.

a. right triangles can be proven congruent by all five congruence theorems.
b. non - right triangles can be proven congruent by hl.
c. right triangles can only be proven congruent by hl.
d. non - right triangles can be proven congruent by all five congruence statements.
e. the acute angles in a right triangle are complementary.
f. if 2 triangles are proven congruent by sas, all corresponding parts are congruent.
g. if two triangles are proven congruent by aas, only 2 pairs of angles are congruent.

Explanation:

Step1: Analyze each statement

• Statement a: Right - triangles can use SSS, SAS, ASA, AAS, and HL. So, they can be proven congruent by all five congruence theorems. This statement is valid.
• Statement b: HL (Hypotenuse - Leg) is a congruence theorem specifically for right - triangles. Non - right triangles cannot be proven congruent by HL. This statement is invalid.
• Statement c: Right - triangles can be proven congruent by SSS, SAS, ASA, AAS, and HL. So, they are not only proven congruent by HL. This statement is invalid.
• Statement d: Non - right triangles can be proven congruent by SSS, SAS, ASA, AAS. They cannot use HL. So, they cannot be proven congruent by all five congruence statements. This statement is invalid.
• Statement e: In a right - triangle, if one angle is \(90^{\circ}\), and the sum of angles in a triangle is \(180^{\circ}\), then the sum of the other two acute angles is \(180^{\circ}- 90^{\circ}=90^{\circ}\). So, the acute angles are complementary. This statement is valid.
• Statement f: By the definition of congruent triangles, if two triangles are congruent (by any congruence criterion like SAS), all corresponding parts (sides and angles) are congruent. This statement is valid.
• Statement g: If two triangles are congruent by AAS (Angle - Angle - Side), then all corresponding parts (three angles and three sides) are congruent. Not just two pairs of angles. This statement is invalid.

Answer:

B. Non - Right triangles can be proven congruent by HL, C. Right triangles can only be proven congruent by HL, D. Non - Right triangles can be proven congruent by all five congruence statements, G. If two triangles are proven congruent by AAS, only 2 pairs of angles are congruent.