Triangle Congruence
The type of triangles required to use the Hypotenuse Leg Theorem
What is
What are right triangles?
Given:
AB ≅ DE
BC ≅ EF
∠B ≅ ∠E
Prove: △ABC ≅ △DEF
Which triangle congruence theorem completes this simple proof?
Answer: SAS, because two pairs of corresponding sides and the included angles are congruent.
A segment begins at a vertex of a triangle and connects to the midpoint of the opposite side.
What type of special segment is described?
Answer: Median
In △ABC, point D is on AC.
It is given that:
AB ≅ BC
AD ≅ DC
BD ≅ BD
Which theorem proves that △ABD ≅ △CBD?
A. ASA
B. AAS
C. SAS
D. SSS
Answer: D. SSS
In △ABC and △DEF:
AB ≅ DE
BC ≅ EF
∠B ≅ ∠E
Which triangle congruence theorem proves that △ABC ≅ △DEF? Explain how you know.
What is
SAS, because two pairs of corresponding sides and the included angles are congruent.
Complete the missing reason:
Statements Reasons
1. AB ≅ DE Given
2. BC ≅ EF Given
3. AC ≅ DF Given
4. △ABC ≅ △DEF ________
Answer: SSS Congruence Theorem
A segment begins at a vertex of a triangle and is perpendicular to the opposite side.
Answer: Altitude, because it begins at a vertex and forms a right angle with the opposite side or its extension.
Given:
△ABC ≅ △DEF
Which statement must be true?
A. AB ≅ EF
B. ∠A ≅ ∠F
C. BC ≅ EF
D. AC ≅ DE
Answer: C. BC ≅ EF
In △JKL and △MNP:
∠J ≅ ∠M
∠L ≅ ∠P
JK ≅ MN
Which triangle congruence theorem proves that △JKL ≅ △MNP? Explain how you know.
What is
AAS, because two pairs of corresponding angles and a non-included side are congruent.
Complete the missing statement:
Statements Reasons
1. △ABC ≅ △DEF Given
2. __________ CPCTC
Prove: ∠C ≅ ∠F
Answer: ∠C ≅ ∠F
A segment divides an angle of a triangle into two angles.
One angle measures:
3x + 5 degrees
The other angle measures:
5x − 15 degrees
If the segment is an angle bisector, find the value of x.
Answer:
Since an angle bisector creates two congruent angles:
3x + 5 = 5x − 15
20 = 2x
x = 10
In △ABC, AD is an angle bisector of ∠A.
If:
m∠BAD = 3x + 7
m∠DAC = 5x − 9
What is the value of x?
A. 4
B. 6
C. 8
D. 12
Answer: C. 8
Since an angle bisector divides an angle into two congruent angles:
3x + 7 = 5x − 9
16 = 2x
x = 8
A student says that two triangles are congruent because they have:
-Two pairs of corresponding congruent sides
-One pair of corresponding congruent angles
The student concludes that the triangles are congruent by SAS.
Is the student always correct? Explain.
What is
No. The congruent angle must be the included angle between the two congruent sides to use SAS. If the angle is not included, the information may represent SSA, which is not a valid triangle congruence theorem.
Given:
AB ≅ DB
∠ABC ≅ ∠DBC
Prove: △ABC ≅ △DBC
Complete the missing statement and identify the congruence theorem.
Statements Reasons
1. AB ≅ DB Given
2. ∠ABC ≅ ∠DBC Given
3. __________ Reflexive Property
4. △ABC ≅ △DBC ___________
Answers:
3. BC ≅ BC
4. SAS Congruence Theorem
A line:
-Passes through the midpoint of a side
-Forms a right angle with that side
-Does not begin at a vertex of the triangle
Which special segment is described? Explain how the clues support your answer.
Answer: Perpendicular bisector. It passes through the midpoint of a side and forms a right angle with the side. Unlike a median or altitude, it does not have to begin at a vertex.
In △ABC and △DEF:
After proving △ABC ≅ △DEF, which statement can be proven using CPCTC?
A. ∠A ≅ ∠F
B. AC ≅ DF
C. AB ≅ EF
D. ∠C ≅ ∠D
Answer: B. AC ≅ DF
The triangles are first proven congruent by SAS, and then AC ≅ DF by CPCTC.
△ABC and △DEF are right triangles.
It is given that:
∠B and ∠E are right angles
AC ≅ DF
AB ≅ DE
2. Explain how you identified the hypotenuses.
What is
HL proves the triangles congruent. AC and DF are the hypotenuses because they are opposite the right angles. AB and DE are corresponding legs.
Given:
AB ≅ DE
BC ≅ EF
∠B ≅ ∠E
Prove: AC ≅ DF
Complete the missing reasons:
Statements Reasons
1. AB ≅ DE Given
2. BC ≅ EF Given
3. ∠B ≅ ∠E Given
4. △ABC ≅ △DEF __________
5. AC ≅ DF __________
Answers:
1. SAS Congruence Theorem
2. CPCTC
A student says:
“Every median is also an altitude because both begin at a vertex and connect to the opposite side.”
Is the student correct? Explain the difference between a median and an altitude. Under what condition could a segment be both?
Answer: No. A median connects a vertex to the midpoint of the opposite side. An altitude begins at a vertex and is perpendicular to the opposite side or its extension.
A segment can be both a median and an altitude when it connects a vertex to the midpoint of the opposite side and forms a right angle with that side, such as the segment drawn from the vertex of an isosceles triangle to its base.
In △ABC and △DEF:
AB ≅ DE
∠A ≅ ∠D
∠B ≅ ∠E
The measures of the corresponding sides are:
AC = 3x + 4
DF = 5x − 10
Answer:
The triangles are congruent by ASA because two pairs of corresponding angles and the included sides are congruent.
Since AC and DF are corresponding sides:
3x + 4 = 5x − 10
14 = 2x
x = 7
Substitute:
AC = 3(7) + 4
AC = 25 units
Answer:
The triangles are congruent by ASA because two pairs of corresponding angles and the included sides are congruent.
Since AC and DF are corresponding sides:
3x + 4 = 5x − 10
14 = 2x
x = 7
Substitute:
AC = 3(7) + 4
AC = 25 units