Presentation on theme: "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."— Presentation transcript:
1Congruent TrianglesIn 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.
2Congruent TrianglesWhich triangles are congruent by the SSS Postulate?Not congruent by SSS
3Congruent Triangles Are these congruent by SAS? Are these congruent by HL?How about:Not right triangles!
4Congruent Triangles Are these triangles congruent by ASA? Yes! No
5Congruent Triangles Are these triangles congruent by AAS? How about now?YesNo
6Congruent TrianglesThere 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.
8Congruent TrianglesA flow proof uses arrows to show the flow of a logical argument.Just like a flow chart does!
9Congruent 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.
11Triangle Relationships InequalityThe longest side and largest angle are opposite each other.The shortest side and smallest angle are opposite each other.
12Triangle Relationships SOLUTIONDraw 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.
13Triangle Relationships InequalityIs it possible to construct a triangle with the given lengths?3, 5, 95+9 > 33+9 > 55+3 > 9 Does not work!Not Possible____________________________________________________________________________________Is it possible to construct a triangle with the given lengths?6, 8, 106+8 > 108+10 > 66+10 > 8It is Possible!
14Triangle 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+=soand
18Triangle Relationships Perpendicular bisectorsDoes 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
19Triangle Relationships Where is the point of concurrency in this triangle?Point GWhat 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!
21Triangle Relationships Angle BisectorsWhat is the angle bisector?FHHow do you know?The Angle Bisectors Theorem
22Triangle 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?SOLUTIONThe 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.
23Triangle Relationships MediansWhat are the medians?BG, CE, AFWhere is the centroid?At DWhat is the length of DG?DG = 6
24Triangle Relationships AltitudesFun Fact: The orthocenter likes to travel!