Question

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4.
Given: KH≅KJ,KM bisectsHJ .
Prove: ∠H ≅ ∠J

Supply the missing reason in Statement 6 of the proof of the Isosceles Triangle Theorem.

Begin with isosceles ∆HKJ with KH≅KJ . Construct KM , a bisector of the base HJ.

A)SSS Postulate
B)Definition of segment bisector
C)Reflexive Property of Congruence
D) CPCTC

Answer 1

The missing reason in Statement 6 of the proof of the Isosceles Triangle Theorem is: D) CPCTC.

In Mathematics and Euclidean Geometry, an isosceles triangle is a type of triangle with two (2) sides that are equal in length and two (2) equal angles.

In Mathematics, CPCTC is an abbreviation for corresponding parts of congruent triangles are congruent and it states that the corresponding angles and side lengths of two (2) or more triangles are congruent if they are both congruent.

In this context, a two-column proof to prove that angles H and J are congruent should be completed as follows;

Statement Reason________________

1. KM bisects HJ. 1. Given

2. HM ≅ JM 2. Definition of segment bisector

3. KH ≅ KJ 3. Given

4. KM KM 4. Reflexive Property of Congruence

5. KHMKJM 5. Side-Side-Side (SSS) Postulate

6. ∠H ≅ ∠J 6. CPCTC

Complete Question:

Given: KH≅KJ,KM bisectsHJ .

Prove: ∠H ≅ ∠J

Supply the missing reason in Statement 6 of the proof of the Isosceles Triangle Theorem.

Begin with isosceles ∆HKJ with KH≅KJ . Construct KM , a bisector of the base HJ.

A)SSS Postulate

B)Definition of segment bisector

C)Reflexive Property of Congruence

D) CPCTC

Answer 2

Option D is correct

CPCT

Explanation:

Given: In an isosceles triangle ΔHKJ with
`KH \cong KJ`

Construct KM, a bisector of the base HJ.

to prove:
`\angle H \cong \angle J`

In ΔKHM and ΔKJM

`\overline{KM}` bisects
`\overline{HJ}` [Given]

Segment bisectors states that a line or segment which cuts another line segment into two equal parts.

then, by definition of Segment bisector :

`\overline{HM} \cong \overline{JM}`

`KH \cong KJ` [Given]

Reflexive property of congruence that any geometric figure is congruent to itself.

`\overline{KM} \cong \overline{KM}` [by definition of Reflexive property of congruence]

SSS(Side-Side-Side) Postulates states that if three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent.

therefore, by SSS postulates

ΔKHM
`\cong` ΔKJM

By CPCT [Corresponding Part of congruent Triangle]

`\angle H \cong \angle J` proved!