Geometry Chapter 44.4/4.5/4.6 – PROVING TRIANGLES CONGRUENT
Warm-up
Write a congruence statement for the triangles given.
M
N O PR
Q
Proving Triangles Congruent
Objective: Students will be able to prove that triangles are
congruent through the use of postulates and theorems.
Agenda
Postulates for Congruent Triangles
Theorems for Congruent Triangles
Identify Congruent Trianlges
Postulates for Congruent Triangles
Postulate 19 – Side-Side-Side Congruence (SSS): If three sides of one
triangle are congruent to three sides of a second triangle, then the
two triangles are congruent.
If 𝑨𝑩 ≅ 𝑭𝑮, 𝑩𝑪 ≅ 𝑮𝑯, and 𝑨𝑪 ≅ 𝑭𝑯
Then ∆𝑨𝑩𝑪 ≅ ∆𝑭𝑮𝑯
Postulates for Congruent Triangles
Postulate 20 – Side-Angle-Side Congruence (SAS): If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent.
If 𝑨𝑩 ≅ 𝑱𝑲, < 𝐀 ≅< 𝑱, and 𝑨𝑪 ≅ 𝑱𝑳
Then ∆𝑨𝑩𝑪 ≅ ∆𝑱𝑲𝑳
Postulates for Congruent Triangles
Postulate 21 – Angle-Side-Angle Congruence (ASA): If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the two triangles are congruent.
If < 𝐀 ≅< 𝑴, 𝑨𝑪 ≅ 𝑴𝑶, and < 𝑪 ≅ < 𝑶
Then ∆𝑨𝑩𝑪 ≅ ∆𝑱𝑲𝑳
Practice with Postulates
State which postulate would be used to prove that these triangles are
congruent. Then write a congruence statement.
𝑴
𝑳
𝑲𝑺
𝑹
𝑻
Practice with Postulates
State which postulate would be used to prove that these triangles are
congruent. Then write a congruence statement.
By SAS
𝑴
𝑳
𝑲𝑺
𝑹
𝑻
Practice with Postulates
State which postulate would be used to prove that these triangles are
congruent. Then write a congruence statement.
By SAS
∆𝑴𝑳𝑲 ≅ ∆𝑻𝑹𝑺
𝑴
𝑳
𝑲𝑺
𝑹
𝑻
Practice with Postulates
State which postulate would be used to prove that these triangles are
congruent. Then write a congruence statement.
𝑾
𝑿
𝒀
𝒁
Practice with Postulates
State which postulate would be used to prove that these triangles are
congruent. Then write a congruence statement.
By SSS
∆𝑾𝑿𝒀 ≅ ∆𝒁𝒀𝑿
𝑾
𝑿
𝒀
𝒁
Practice with Postulates
State which postulate would be used to prove that these triangles are
congruent. Then write a congruence statement.
By SAS
∆𝑨𝑩𝑪 ≅ ∆𝑫𝑭𝑪
𝑨
𝑩
𝑪
𝑫
𝑭
Right Triangles
The sides of a right triangle named as such:
The side opposite the right angle is known as the Hypotenuse
The other two sides are known as the Legs
Hypotenuse
Leg
Leg
Theorems for Congruent Triangles
Theorem 4.5 – Hypotenuse-Leg Congruence (HL): If the hypotenuse
and a leg of a right triangle are congruent to a hypotenuse and a
leg of a second right triangle, then the two triangles are congruent.
If 𝑨𝑩 ≅ 𝑫𝑬, 𝑩𝑪 ≅ 𝑬𝑭, and < 𝑪 and < 𝑭are right angles,
Then ∆𝑨𝑩𝑪 ≅ ∆𝑫𝑬𝑭
Theorems for Congruent Triangles
Theorem 4.6 – Angle-Angle-Side Congruence (AAS): If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the two triangles are congruent.
If < 𝑩 ≅< 𝑬, < 𝐂 ≅ < 𝑭, and 𝑨𝑪 ≅ 𝑫𝑭
Then ∆𝑨𝑩𝑪 ≅ ∆𝑫𝑬𝑭
Practice with Theorems
State which theorem would be used to prove that these triangles are
congruent. Then write a congruence statement.
𝑭
𝑮 𝑯
𝑼
𝑻
𝑽
Practice with Theorems
State which theorem would be used to prove that these triangles are
congruent. Then write a congruence statement.
By AAS
∆𝑭𝑮𝑯 ≅ ∆𝑽𝑻𝑼
𝑭
𝑮 𝑯
𝑼
𝑻
𝑽
Practice with Theorems
State which theorem would be used to prove that these triangles are
congruent. Then write a congruence statement.
𝑨
𝑪
𝑩
𝑫
By HL
∆𝑨𝑩𝑫 ≅ ∆𝑨𝑪𝑫
Practice with Theorems
State which theorem would be used to prove that these triangles are
congruent. Then write a congruence statement.
𝑱
𝑯 𝑰
𝑲
𝑮
Practice with Theorems
State which theorem would be used to prove that these triangles are
congruent. Then write a congruence statement.
𝑱
𝑯
𝑮
𝑰
K
𝑮
How was this made?
Practice with Theorems
State which theorem would be used to prove that these triangles are
congruent. Then write a congruence statement.
𝑱
𝑯
𝑮
𝑰
K
𝑮
By AAS
∆𝑮𝑯𝑱 ≅ ∆𝑮𝑰𝑲
