Geometry Triangle Congruence Theorems Congruent Triangles z Congruent

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Geometry Triangle Congruence Theorems

Geometry Triangle Congruence Theorems

Congruent Triangles z. Congruent triangles have three congruent sides and three congruent angles. z.

Congruent Triangles z. Congruent triangles have three congruent sides and three congruent angles. z. However, triangles can be proved congruent without showing 3 pairs of congruent sides and angles.

The Triangle Congruence Postulates &Theorems FOR ALL TRIANGLES SSS SAS ASA FOR RIGHT TRIANGLES

The Triangle Congruence Postulates &Theorems FOR ALL TRIANGLES SSS SAS ASA FOR RIGHT TRIANGLES ONLY HL AAS

Theorem z. If two angles in one triangle are congruent to two angles in

Theorem z. If two angles in one triangle are congruent to two angles in another triangle, the third angles must also be congruent. z. Think about it… they have to add up to 180°.

A closer look. . . z If two triangles have two pairs of angles

A closer look. . . z If two triangles have two pairs of angles congruent, then their third pair of angles is congruent. z But do the two triangles have to be congruent? 85° 30°

Example 30° Why aren’t these triangles congruent? What do we call these triangles?

Example 30° Why aren’t these triangles congruent? What do we call these triangles?

z. So, how do we prove that two triangles really are congruent?

z. So, how do we prove that two triangles really are congruent?

ASA (Angle, Side, Angle) A C D z. If two angles and the included

ASA (Angle, Side, Angle) A C D z. If two angles and the included side of one triangle are congruent to two angles and the B included side of another triangle, . . . F E then the 2 triangles are CONGRUENT!

AAS (Angle, Side) Special case of ASA z If two angles and a nonincluded

AAS (Angle, Side) Special case of ASA z If two angles and a nonincluded side of one triangle are congruent to two angles and the corresponding non- C included side of another triangle, . . . then the 2 triangles are CONGRUENT! A D B F E

SAS (Side, Angle, Side) A C D z. If in two triangles, two sides

SAS (Side, Angle, Side) A C D z. If in two triangles, two sides and the included angle of one are congruent to two sides B and the included angle of the other, . . . F E then the 2 triangles are CONGRUENT!

SSS (Side, Side) A C B D F E z. In two triangles, if

SSS (Side, Side) A C B D F E z. In two triangles, if 3 sides of one are congruent to three sides of the other, . . . then the 2 triangles are CONGRUENT!

HL (Hypotenuse, Leg) z If both hypotenuses and a pair of legs of two

HL (Hypotenuse, Leg) z If both hypotenuses and a pair of legs of two RIGHT triangles are congruent, . . . then the 2 triangles are CONGRUENT! A C B D F E

Example 1 A C B D E F z Given the markings on the

Example 1 A C B D E F z Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson?

Example 2 A C B D E F z Given the markings on the

Example 2 A C B D E F z Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson?

Example 3 A C B D z Given the markings on the diagram, is

Example 3 A C B D z Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson?

D Example 4 z Why are the two triangles congruent? SAS F E A

D Example 4 z Why are the two triangles congruent? SAS F E A z What are the corresponding vertices? A D C E C B B F

Example 5 A z Why are the two triangles congruent? SSS D B z

Example 5 A z Why are the two triangles congruent? SSS D B z What are the corresponding A C vertices? ADB CDB ABD C

Example 6 B C z. Given: A Are the triangles congruent? Why? S S

Example 6 B C z. Given: A Are the triangles congruent? Why? S S S D

Example 7 Q P z. Given: T m QSR = m PRS = 90°

Example 7 Q P z. Given: T m QSR = m PRS = 90° R S n Are the Triangles Congruent? Why? QSR PRS = 90° R H S

Summary: ASA - Pairs of congruent sides contained between two congruent angles AAS –

Summary: ASA - Pairs of congruent sides contained between two congruent angles AAS – Pairs of congruent angles and the side not contained between them. SAS - Pairs of congruent angles contained between two congruent sides SSS - Three pairs of congruent sides

Summary --for Right Triangles Only: HL – Pair of sides including the Hypotenuse and

Summary --for Right Triangles Only: HL – Pair of sides including the Hypotenuse and one Leg

THE END!!!

THE END!!!

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