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
Wait, #4 is SSS Postulate, right?
Anyone, please?
Correct.
I hope.
1. To prove that ∠NLO ≅ ∠NPM, we need to show that triangle LNO is congruent to triangle PNM.
Hector has already established that segment LN is congruent to segment NP and ∠1 ≅ ∠2.
To complete the proof, Hector can use the Angle-Angle-Side Postulate.
This postulate states that if two angles of one triangle are congruent to two angles of another triangle, and the lengths of the sides included between the congruent angles are also congruent, then the triangles are congruent.
In this case, ∠N ≅ ∠N (Reflexive Property) and segment LO is congruent to segment MP (given). Therefore, Hector can use the Angle-Angle-Side Postulate to prove that triangle LNO is congruent to triangle PNM, which implies that ∠NLO ≅ ∠NPM.
2. To prove that ΔBJI ~ ΔCJG, we need to establish the necessary conditions for similarity.
The correct choice is ∠BIJ ≅ ∠CGJ and ∠BJI ≅ ∠CJG.
For two triangles to be similar, the corresponding angles need to be congruent, and the corresponding sides can be proportional but not necessarily congruent. In this case, the given conditions satisfy the angle congruence requirement. ∠BIJ ≅ ∠CGJ and ∠BJI ≅ ∠CJG indicate that the angles in the triangles are congruent, which is sufficient to prove similarity.
3. Abdul can use the Corresponding Angles Theorem to determine that certain angles are congruent.
The Corresponding Angles Theorem states that if two parallel lines are intersected by a transversal, then the corresponding angles formed are congruent.
In this case, Abdul's neighborhood map depicts intersecting streets forming a triangle. The triangle is intersected by a line formed by the elementary and middle school. According to the given information, the angle relationship between the elementary school, middle school, and Abdul's home is congruent to the angle relationship between Abdul's friend's house, the high school, and Abdul's home. These angles are corresponding angles as they are formed by the intersection of two parallel lines (the streets) and a transversal (the line formed by the schools). Therefore, Abdul can use the Corresponding Angles Theorem to determine that certain angles are congruent.
4. In statement 5, to prove that triangle NRM is congruent to triangle EFD, the best reasoning is the SAS Postulate.
The SAS (Side-Angle-Side) Postulate states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.
In this case, Ben has already shown that r2 = m2 + n2 and f2 = m2 + n2 (given and Pythagorean Theorem). By substitution (statement 3), he deduced f2 = r2. Then, using the Square Root Property of Equality (statement 4), he found f = r.
Now, to prove that triangle NRM is congruent to triangle EFD, Ben can use the SAS Postulate. He has shown that side NR (length r) is congruent to side EF (length f), side RM (length m) is congruent to side ED (length m), and the included angle NRM is the same as the included angle EFD (since Ben constructed EFD to be a right triangle with the same leg lengths m and n). Therefore, by the SAS Postulate, he can conclude that triangle NRM is congruent to triangle EFD, which implies that angle NRM is a right angle (CPCTC) and triangle NRM is a right triangle.