Warm-up 3/191) Sketch ΔABC and ΔRDH so that they are

congruent triangles.2) What things must we know to say that two

polygons are congruent?

Announcements 3/191) Test needs to be made up by Friday.2) Homework 6.1 due Monday.3) New Bathroom passes.4) Ways to raise your grade.5) Participation

Chapter 6 Congruent Triangles and Quadrilaterals.

6.1 Congruence Postulates and Theorems

Recall:Congruent polygons are polygons that have three properties:1) Same number of sides.2) Corresponding sides are congruent.3) Corresponding angles are congruent.So to determine if two triangles are congruent you would have to show that all three corresponding sides are congruent and all three corresponding angles are congruent.Objective: Today we will learn that only three of these six congruence's are needed to determine if triangles are congruent.

Ex #1Given: △SPQ and △RPQ;Prove: △SPQ ≅△RPQThink about what things you must show in order to prove that the two triangles are congruent.Pair with a partner and determine what you must show.Share

PRQPSQQPS;QPR;RPSQ;RPSP

Ex #1Now lets try the proof.

NOTE: The reflexive propertystates that a side or angle isCongruent to itself.

Statements ReasonsStatements Reasons

1)△SPQ and △RPQ 1) Given

Statements Reasons

1)△SPQ and △RPQ 1) Given

2) 2) Theorem 5.11

Statements Reasons

1)△SPQ and △RPQ 1) Given

2) 2) Theorem 5.11

3) 3) Reflexive Property

4) △SPQ ≅△RPQ 4) Definition of congruent polygons

PRQPSQQPS;QPR;RPSQ;RPSP

QPSQPR

QPPQ

PRQPSQQPS;QPR;RPSQ;RPSP PRQPSQQPS;QPR;RPSQ;RPSP

Statements Reasons

1)△SPQ and △RPQ 1) Given

2) 2) Theorem 5.11

3) 3) Reflexive Property

QPSQPR

Statements Reasons

1)△SPQ and △RPQ 1) Given

2) 2) Theorem 5.11

3) 3) Reflexive Property

4) △SPQ ≅△RPQ 4) Definition of congruent polygons

QPSQPR

QPPQ

PRQPSQQPS;QPR;RPSQ;RPSP

We will now try to discover what three congruence's we need to be able to tell if two triangles are congruent.First let us draw two angles that are congruent.Second make one side of each angle congruent line segments.Third make the second sideof each angle congruent.Now how many possible tri-angles can be formed?Only one!

Thus if we know two sides and the angle that is formed by these sides are congruent then we know that the two triangles are congruent.This leads us to a Postulate. That means we cannot prove it but we can “see” that this statement must be true.Postulate 6.1 Side-Angle-Side Congruence Postulate. If two sides and an included angle of one triangle are congruent to the corresponding two sides and included angle of another triangle, then the two triangles are congruent.

Lets try anotherFirst let us draw two segments that are congruent.Second, using this segment as one side of an angle draw two congruent angles.Third, using the original segment draw two more congruent angles from the other vertex.Now how many possible tri-angles can be formed?Only one!

Thus if we know two angles and the side that is between these angles are congruent then we know that the two triangles are congruent.This leads us to a Postulate. That means we cannot prove it but we can “see” that this statement must be true.Postulate 6.2 Angle-Side-Angle Congruence Postulate. If two angles and an included side of one triangle are congruent to the corresponding two angles and included side of another triangle, then the two triangles are congruent.

Warm-up 3/201) Use Postulates 6.1 and 6.2 to state whether the

triangles are congruent. State which postulate you would use.

Announcements 3/191) Test needs to be made up by Friday.2) Homework 6.1 due Monday.3) New Bathroom passes.4) Participation

Ex #2Given: AD ∥BC; AB DC∥Prove: △ABD ≅△CDBThink about what things you must show in order to prove that the two triangles are congruent.Pair with a partner and determine what you must show.Share

Theorem 6.1 Side-Angle-Angle congruence theorem. If two angles of a triangle and a side opposite one of the two angles are congruent to the corresponding angles and side of another triangle, then the two triangles are congruent.Theorem 6.2 Isosceles Triangle Theorem. In an isosceles triangle the base angles are congruent.Theorem 6.3 If two angles of a triangle are congruent, then the sides opposite those angles are congruent, and the triangle is an isosceles triangle.Theorem 6.4 A triangle is equilateral if and only if it is equiangular.

Warm-up 3/231) Finish the proof

Statements Reasons

1) Bisects 1)

2) 2) Definition of angle bisector

3) 3)

4) 4)

5) 5)

LN MNOandMLO

LNOLNMNLOMLN ;

LNLN

OLNMLN

OM

L

NM O

Announcements 3/231) Homework 6.1 due Today.2) Homework 6.1 Part 2 due Wednesday.

Cont of 6.1 Congruence Postulates and Theorems

Review:So what Triangle congruence Postulates and

theorems do we have so far?-SAS-ASA-SAA

That leaves us with 3 more possibilities:-AAA-SSS-ASS

Objective: Today we will find one more triangle congruence theorem and will understand why the other options do not give us congruence theorems.SSSFirst determine the 3 side lengths you will have to make your two triangles. Now how many possible tri-angles can be formed?Only one!Theorem 6.5 side-Side-Side Congruence Theorem. If each side of one triangle is congruent to the corresponding side of a second triangle then the two triangles are congruent.

AAAFirst determine the 3 angle measures you will have to make your two triangles. Now how many possible tri-angles can be formed?Infinitely many!So if you know that two triangles have 3 angles that are congruent you CANNOT say that the triangles are congruent.

ASSFirst draw an acute angle.Second choose two lengths for your sides and make one ray of your angle one of these lengths.Now how many possible tri-angles can be formed?2!So if you know that two triangles have an angle, a side included in the angle, and a side opposite the angle that are congruent(ASS) you CANNOT say that the triangles are congruent.

Warm-up 3/24

Announcements 3/241) Homework 6.1 Part 2 due Wednesday.2) CM #29.1 (Triangle congruence's) will be on

Wednesday.3) Unit Quiz next Friday the 24th

4) Period 3, Bring socks.5) Homework Questions????