Congruent Triangles In two congruent figures, all the parts of one figure are congruent to the corresponding parts of the other figure. This means there.

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Presentation transcript:

Congruent Triangles In two congruent figures, all the parts of one figure are congruent to the corresponding parts of the other figure. This means there will be corresponding sides that are congruent. There will also be corresponding angles that are congruent. What are the corresponding sides? What are the corresponding angles? A coordinate proof involves placing geometric figures in a coordinate plane.

Congruent Triangles Which triangles are congruent by the SSS Postulate? Not congruent by SSS

Congruent Triangles Are these congruent by SAS? Are these congruent by HL? How about: Not right triangles!

Congruent Triangles Are these triangles congruent by ASA? Yes! No

Congruent Triangles Are these triangles congruent by AAS? How about now? Yes No

Congruent Triangles There are a couple of methods for organizing your thoughts when proving triangle congruency. The first is to use a two-column proof. The second is to use a flow proof.

Congruent Triangles Given: Prove:

Congruent Triangles A flow proof uses arrows to show the flow of a logical argument. Just like a flow chart does!

Congruent Triangles So remember: SSS, SAS, ASA, AAS Postulates and the HL Theorem will help everyone to be congruent. But be careful during a test! Make sure you don’t need to call AAA to bail out your SSA.

Congruent Triangles Problems Pg 239 #15-16 Pg 245 # 17-18 Pg 247 #9-10

Triangle Relationships Inequality The longest side and largest angle are opposite each other. The shortest side and smallest angle are opposite each other.

Triangle Relationships SOLUTION Draw a diagram and label the side lengths. The peak angle is opposite the longest side so, by Theorem 5.10, the peak angle is the largest angle.

Triangle Relationships Inequality Is it possible to construct a triangle with the given lengths? 3, 5, 9 5+9 > 3 3+9 > 5 5+3 > 9  Does not work! Not Possible ____________________________________________________________________________________ Is it possible to construct a triangle with the given lengths? 6, 8, 10 6+8 > 10 8+10 > 6 6+10 > 8 It is Possible!

Triangle Relationships What can we say about angle 1? Think about it: The angles of a triangle have to sum to 180. The angles that form a line must sum to 180. Thus + = so and

Triangle Relationships Hinge Theorem

Triangle Relationships Hinge Theorem CD, BC, BD, AB, AD 60 35

Triangle Relationships Problems Pg 287 #1-10, 15-28 Pg 294 #1-13

Triangle Relationships Perpendicular bisectors Does this triangle have a perpendicular bisector? How do you know? Theorem 5.3 proves D is on the perpendicular bisector and BD makes a right angle with AC at its midpoint. Yes, segment BD

Triangle Relationships Where is the point of concurrency in this triangle? Point G What special type of point of concurrency is this? It is a circumcenter. What is special about the red lines? They are congruent. **Note: The circumcenter can be outside of the triangle if you have an obtuse triangle!

Triangle Relationships Problems Math I Pg 266 #1-18 Pg 268 #1-9

Triangle Relationships Angle Bisectors What is the angle bisector? FH How do you know? The Angle Bisectors Theorem

Triangle Relationships A soccer goalie’s position relative to the ball and goalposts forms congruent angles, as shown. Will the goalie have to move farther to block a shot toward the right goalpost R or the left goalpost L? SOLUTION The congruent angles tell you that the goalie is on the bisector of LBR. By the Angle Bisector Theorem, the goalie is equidistant from BR and BL . So, the goalie must move the same distance to block either shot.

Triangle Relationships Medians What are the medians? BG, CE, AF Where is the centroid? At D What is the length of DG? DG = 6

Triangle Relationships Altitudes Fun Fact: The orthocenter likes to travel!

Triangle Relationships Problems Math I Pg 274 #1-12, 14-17 Pg 280-281 #1-20 Pg 282 #1-5