Download Study each pair of right triangles. abc 1. Is each pair

Extend 4-5 Geometry Lab: Congruence in Right Triangles
Study each pair of right triangles.
a.
b.
c.
1. Is each pair of triangles congruent? If so, which congruence theorem or postulate applies?
SOLUTION: a. Two pairs of corresponding sides and one included angle are congruent. Therefore, by SAS postulate both
triangles are congruent.
b. Two pairs of corresponding angles and one non-included side are congruent. By AAS postulate the triangles are
congruent.
c. Two pairs of corresponding angles and one included side are congruent in the triangles. So, the triangles are
congruent by using ASA postulate.
2. Rewrite the congruence rules from Exercise 1 using leg, (L), or hypotenuse, (H), to replace side. Omit the A for
any right angle since we know that all right triangles contain a right angle and all right angles are congruent.
SOLUTION: a. Two pairs of corresponding legs of the right triangles are congruent. By Leg-Leg (LL) congruence, the triangles
are congruent.
b. The hypotenuses and a pair of corresponding angles of the right triangles are congruent. So, the triangles are
congruent by Hypotenuse-Angle (HA) congruence theorem.
c. The pair of corresponding acute angles and the pair of corresponding legs are congruent. Therefore, the triangles
are congruent by Leg-Angle (LA) congruence.
3. MAKE A CONJECTURE If you know that the corresponding legs of two right triangles are congruent, what
other information do you need to declare the triangles congruent? Explain.
SOLUTION: If the two legs are congruent then the included angle is the right angle. The two triangles would be congruent by the
SAS postulate. Which corresponds to the Leg-Leg (LL) congruence. None; two pairs of legs congruent is sufficient
for proving right triangles congruent using Leg-Leg (LL) congruence.
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SOLUTION: If the two legs are congruent then the included angle is the right angle. The two triangles would be congruent by the
SAS4-5
postulate.
Which
corresponds
to theinLeg-Leg
(LL) congruence. None; two pairs of legs congruent is sufficient
Extend
Geometry
Lab:
Congruence
Right Triangles
for proving right triangles congruent using Leg-Leg (LL) congruence.
4. Does the model yield a unique triangle?
SOLUTION: All three sides of the triangle are specified. Yes, the triangle is unique.
5. Can you use the lengths of the hypotenuse and a leg to show right triangles are congruent?
SOLUTION: Yes. Hypotenuse-Leg(HL) congruence can be applied here.
6. Make a conjecture about the case of SSA that exists for right triangles.
SOLUTION: After performing the activity, knowing the length of the hypotenuse and a leg will yield the same triangle every time.
SSA or HL-hypotenuse leg is a valid test of congruence for right triangles.
Determine whether each pair of triangles is congruent. If yes, tell which postulate or theorem applies.
7. SOLUTION: eSolutions
by Cogneroangles
TwoManual
pairs -ofPowered
corresponding
and a pair of corresponding legs are congruent. Therefore, the triangles are
congruent by Leg-Angle (LA) congruence.
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6. Make a conjecture about the case of SSA that exists for right triangles.
SOLUTION: After4-5
performing
theLab:
activity,
knowing the
the hypotenuse and a leg will yield the same triangle every time.
Extend
Geometry
Congruence
in length
Right of
Triangles
SSA or HL-hypotenuse leg is a valid test of congruence for right triangles.
Determine whether each pair of triangles is congruent. If yes, tell which postulate or theorem applies.
7. SOLUTION: Two pairs of corresponding angles and a pair of corresponding legs are congruent. Therefore, the triangles are
congruent by Leg-Angle (LA) congruence.
8. SOLUTION: The corresponding angles of the triangles are congruent but no sides are congruent. Therefore, no postulates can be
applied here. The triangles are not congruent.
9. SOLUTION: The hypotenuses of the triangles are congruent and the corresponding legs are also congruent. By Hypotenuse-Leg
(HL) congruence, the triangles are congruent.
PROOF Write a proof for each of the following.
10. Theorem 4.6
SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here,
you are given that both triangles are right and that the legs are congruent. So, you are given two congruent sides and
one congruent angle. Now use what you have learned about triangle congruency to prove that if the legs of one right
triangle are congruent to the corresponding legs of another right triangle, then the triangles are congruent.
Given:
and
are right triangles.
∠E and ∠S are right angles.
Prove:
Proof: We are given that
and
congruent, E S. Therefore, by SAS,
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11. Theorem
SOLUTION: E and
S are right angles. Since all right angles are
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Prove:
Proof: We are given that
and E and S are right angles. Since all right angles are
Extend 4-5 Geometry Lab: Congruence in Right Triangles
congruent, E S. Therefore, by SAS,
11. Theorem 4.7
SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here,
you are given that both triangles are right, their hypotenuses are congruent, and an angle is congruent. So, you are
given two congruent angles and one congruent side. Now use what you have learned about triangle congruency to
prove that if the hypotenuse and acute angle of one right triangle are congruent to the hypotenuse and corresponding
acute angle of another right triangle, then the two triangles are congruent.
Given:
and are right triangles.
A and X are right angles.
Prove:
Proof: We are given that
and are right triangles with right angles Y. Since all right angles are congruent, A X. Therefore,
A and X,
by AAS.
and
B
12. Theorem 4.8 (Hint: There are two possible cases.)
SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here,
you are given that both triangles are right, one of the legs is congruent, and one of the angles is congruent. So, you
are given two congruent angles and one congruent side. Now use what you have learned about triangle congruency
to prove that if one leg and an acute angle of one right triangle are congruent to the corresponding leg and acute
angle of another right triangle, then the triangles are congruent.
Case 1:
Given:
F
and are right triangles. C
Prove:
Proof: It is given that
and are right triangles, By the definition of right triangles,
A and D are right angles. Thus, A
D since all right angles are congruent.
by ASA.
Case 2:
Given:
and are right triangles. Prove:
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Proof: It is given that
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and
are right triangles,
.
Prove:
Proof:
are givenLab:
that Congruence
and in Right
are right triangles with right angles Extend
4-5We
Geometry
Triangles
Y. Since all right angles are congruent, A X. Therefore,
A and X,
by AAS.
and
B
12. Theorem 4.8 (Hint: There are two possible cases.)
SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here,
you are given that both triangles are right, one of the legs is congruent, and one of the angles is congruent. So, you
are given two congruent angles and one congruent side. Now use what you have learned about triangle congruency
to prove that if one leg and an acute angle of one right triangle are congruent to the corresponding leg and acute
angle of another right triangle, then the triangles are congruent.
Case 1:
Given:
F
and are right triangles. C
Prove:
Proof: It is given that
and are right triangles, By the definition of right triangles,
A and D are right angles. Thus, A
D since all right angles are congruent.
by ASA.
Case 2:
Given:
and are right triangles. Prove:
Proof: It is given that
and
By the definition of right triangle,
congruent.
by AAS.
are right triangles,
A and E are right angles. Thus,
A
.
E since all right angles are
13. Theorem 4.9
SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here,
you are given that both triangles are right, their hypotenuses are congruent, and one of their legs is congruent. So,
you are given two congruent sides and one congruent angle. Now use what you have learned about triangle
congruency to prove that if the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and
corresponding leg of another right triangle, then the triangles are congruent.
Given:
are right triangles.
Prove:
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Proof: It is given that
and
are right triangles,
By the definition of right triangle, A and E are right angles. Thus,
congruent.
Extend
4-5 Geometry Lab: Congruence in Right Triangles
by AAS.
A
.
E since all right angles are
13. Theorem 4.9
SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here,
you are given that both triangles are right, their hypotenuses are congruent, and one of their legs is congruent. So,
you are given two congruent sides and one congruent angle. Now use what you have learned about triangle
congruency to prove that if the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and
corresponding leg of another right triangle, then the triangles are congruent.
Given:
are right triangles.
Prove:
Proof:
1.
are right triangles,
2. AB = DE, BC = EF (Def. of )
2
2
2
2
2
2
2
(Given)
2
2
3. (AB) + (CA) = (BC) , (DE) + (FD) = (EF) (Pythagorean Theorem.)
2
2
2
2
4. (AB) + (CA) = (DE) + (FD) (Subs. Prop.)
2
5. (AB) + (CA) = (AB) + (FD) (Subs. Prop.)
2
2
6. (CA) = (FD) (Subt. Prop.)
7. CA = FD (A property of square roots)
8.
(Definition of congruent segment)
9.
(SSS)
Use the figure at the right.
14. Given:
Prove:
SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here,
you are given that both triangles are right, they have a congruent leg, and that their hypotenuses are congruent. So,
you are given two congruent sides and one congruent angle. Now use what you have learned about triangle
congruency to prove that if the legs of one right triangle are congruent to the corresponding legs of another right
triangle, then the triangles are congruent. Finally, use the fact that corresponding parts of congruent figures are
themselves congruent, to prove the line segments congruent.
Proof:
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Statements
(Reasons)
1.
(Given)
2. ∠ABC is a right angle, ∠DCB is a right angle. ( lines from right angles)
5. (AB) + (CA) = (AB) + (FD) (Subs. Prop.)
2
2
6. (CA) = (FD) (Subt. Prop.)
7. CA = FD (A property of square roots)
8. 4-5 Geometry
(Definition
congruent segment)
Extend
Lab: of
Congruence
in Right Triangles
9.
(SSS)
Use the figure at the right.
14. Given:
Prove:
SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here,
you are given that both triangles are right, they have a congruent leg, and that their hypotenuses are congruent. So,
you are given two congruent sides and one congruent angle. Now use what you have learned about triangle
congruency to prove that if the legs of one right triangle are congruent to the corresponding legs of another right
triangle, then the triangles are congruent. Finally, use the fact that corresponding parts of congruent figures are
themselves congruent, to prove the line segments congruent.
Proof:
Statements (Reasons)
1.
(Given)
2. ∠ABC is a right angle, ∠DCB is a right angle. ( lines from right angles)
3.
is a right triangle, is a right triangle. (Definition of right triangles )
4.
(Given)
5.
(Reflection Property of Congruence)
6.
(HL congruence)
7.
(CPCTC)
15. Given:
E is the midpoint of
Prove:
and SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here,
you are given that both lines are parallel, one right angle, and a midpoint. Use what you know about parallel lines and
perpendicularity to prove that both triangles are right. Then use what you know about midpoints to prove segments
congruent. Now use what you know about vertical angles to find another pair of angles congruent. So, you are given
two congruent sides and one congruent angle. Now use what you have learned about triangle congruency to prove
the triangles are congruent. Finally, use the fact that corresponding parts of congruent figures are themselves
congruent, to prove the line segments congruent.
Proof:
Statements (Reasons)
1.
(Given)
2.
(In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other.)
3. ∠ABC is a right angle, ∠DCB is a right angle. ( lines form rt. s.)
4.
is a right triangle,
is a right triangle. (Def of rt )
5. E is the midpoint of
(Given)
and 6.
(Midpoint Thm.)
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7.
(Vertical angles are congruent.)
8.
(SAS)
4.
(Given)
5.
(Reflection Property of Congruence)
6.
(HL congruence)
Extend
4-5
Geometry
Lab: Congruence in Right Triangles
7.
(CPCTC)
15. Given:
E is the midpoint of
Prove:
and SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here,
you are given that both lines are parallel, one right angle, and a midpoint. Use what you know about parallel lines and
perpendicularity to prove that both triangles are right. Then use what you know about midpoints to prove segments
congruent. Now use what you know about vertical angles to find another pair of angles congruent. So, you are given
two congruent sides and one congruent angle. Now use what you have learned about triangle congruency to prove
the triangles are congruent. Finally, use the fact that corresponding parts of congruent figures are themselves
congruent, to prove the line segments congruent.
Proof:
Statements (Reasons)
1.
(Given)
2.
(In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other.)
3. ∠ABC is a right angle, ∠DCB is a right angle. ( lines form rt. s.)
4.
is a right triangle,
is a right triangle. (Def of rt )
5. E is the midpoint of
(Given)
and 6.
(Midpoint Thm.)
7.
(Vertical angles are congruent.)
8.
(SAS)
9.
(CPCTC)
10.
(Reflexive Property of Congruence)
11.
(LL)
12.
(CPCTC)
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