Geometry Chapter 4 … · Postulates for Congruent Triangles Postulate 20 –Side-Angle-Side Congruence (SAS): If two sides and the included angle of one triangle are congruent to

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