Question

Cai tried to prove that \triangle FGH\cong \triangle HIJ△FGH≅△HIJtriangle, F, G, H, \cong, triangle, H, I, J. Statement Reason 1 FG=HI=6FG=HI=6F, G, equals, H, I, equals, 6 Given 2 FH=HJ=4FH=HJ=4F, H, equals, H, J, equals, 4 Given 3 \overline{FG} \parallel \overline{HI} FG ∥ HI start overline, F, G, end overline, \parallel, start overline, H, I, end overline Given 4 \angle HFG\cong\angle JHI∠HFG≅∠JHIangle, H, F, G, \cong, angle, J, H, I When a transversal crosses parallel lines, alternate interior angles are congruent. 5 \triangle FGH\cong \triangle HIJ△FGH≅△HIJtriangle, F, G, H, \cong, triangle, H, I, J Side-angle-side congruence What is the first error Cai made in his proof? Choose 1 answer: Choose 1 answer: (Choice A) A Cai used an invalid reason to justify the congruence of a pair of sides or angles. (Choice B) B Cai only established some of the necessary conditions for a congruence criterion. (Choice C) C Cai established all necessary conditions, but then used an inappropriate congruence criterion. (Choice D) D Cai used a criterion that does not guarantee congruence.

Answer 1

The first error made by Cai in his proof was: the use of an invalid reason to justify the congruence of a pair of sides or angles. (Option A)

Recall:

• If two triangles have two pairs of congruent sides and a pair of included side that are congruent, both triangles are considered congruent by the Side-Angle-Side Congruence Theorem (SAS).

△FGH and △HIJ are shown in the diagram attached below. Also, the diagram shows the proof of Cai.

The first error that can be noted is that Cai the reason given for stating that $\angle HFG \cong \angle JHI$ is false and invalid.

<HFG and <JHI are corresponding angles which makes them congruent. They are not alternate interior angles.

• Therefore, the first error made by Cai in his proof was: the use of an invalid reason to justify the congruence of a pair of sides or angles. (Option A)

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