Geometry

Please help

1. If segment LN is congruent to segment NP and ∠1 ≅ ∠2, prove that ∠NLO ≅ ∠NPM:

Overlapping triangles LNO and PNM. The triangles intersect at point Q on segment LO of triangle LNO and segment MP of triangle PNM.

Hector wrote the following proof for his geometry homework for the given problem.

Statements Reasons
segment LN is congruent to segment NP Given
∠1 ≅ ∠2 Given
∠N ≅ ∠N Reflexive Property
ΔLNO ≅ ΔPNM
∠NLO ≅ ∠NPM Corresponding Parts of Congruent Triangles Are Congruent

Which of the following completes Hector's proof?
Angle-Angle-Side Postulate***
Angle-Side-Angle Postulate
Side-Angle-Side Postulate
Side-Side-Side Postulate

2. Use the figure below to answer the question that follows:

Intersecting triangles ACE and BDF. They intersect at points G, H, I, and J.

What must be given to prove that ΔBJI ~ ΔCJG?

segment BH is congruent to segment CH and segment BG is congruent to segment CI
∠BIJ ≅ ∠CGJ and ∠JBI ≅ ∠JIB
segment BI is congruent to segment CG and segment JI is congruent to segment JG
∠BIJ ≅ ∠CGJ and ∠BJI ≅ ∠CJG***

3. Abdul is making a map of his neighborhood. He knows the following information:

His home, the middle school, and high school are all on the same street.
His home, the elementary school, and his friend's house are on the same street.
The angle between the elementary school, middle school, and his home is congruent to the angle between his friend's house, the high school, and his home.

A street map is shown. The streets form a triangle comprised of the locations of home, friends house, and the high school. The triangle is intersected by a line formed by the elementary and middle school.

What theorem can Abdul use to determine that certain angles are congruent?
Corresponding Angles Theorem***
Vertical Angles Theorem
Pythagorean Theorem
Angle-Angle-Side Theorem

4.The figure below shows triangle NRM with r2 = m2 + n2:

Triangle NRM has legs m and n, and r is the length of its longest side.

Ben constructed a right triangle EFD with legs m and n, as shown below:

Triangle EFD has legs m and n and hypotenuse f.

He made the following table to prove that triangle NRM is a right triangle:

Statement Reason
1. r2 = m2 + n2 Given
2. f2 = m2 + n2 Pythagorean Theorem
3. f2 = r2 Substitution
4. f = r Square Root Property of Equality
5. Triangle NRM is congruent to triangle EFD ?
6. Angle NRM is a right angle CPCTC
7. Triangle NRM is a right triangle Angle NRM is a right angle

Which reason best fits statement 5?
SSS Postulate
SAS Postulate***
AAA Postulate
AAS Postulate

  1. 👍
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  3. 👁
  1. Wait, #4 is SSS Postulate, right?

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    2. 👎
  2. Anyone, please?

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    2. 👎
  3. Correct.

    I hope.

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