Geometry 3 4 Triangle Congruence Theorems Congruent Triangles

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

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

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

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

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?

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

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

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

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

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

HA (Hypotenuse, Angle) A z If both hypotenuses and a pair of acute angles

LA (Leg, Angle) A z If both hypotenuses and a pair of acute angles

LL (Leg, Leg) A z If both pair of legs of two RIGHT triangles

Example 1 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

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

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

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

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

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

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

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

Slides: 25

Download presentation

Geometry 3. 4 Triangle Congruence Theorems

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 ASA SAS FOR RIGHT TRIANGLES ONLY HL AAS

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 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?

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 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 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 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 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 RIGHT triangles are congruent, . . . A C B D F E

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

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

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

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 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 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 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 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 S D

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 – 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 one Leg HA – Pair of hypotenuses and one acute angle LL – Both pair of legs LA – One pair of legs and one pair of acute angles

THE END!!!