Geometry CP Unit 4 (Congruency of Triangles) Notes S
4.1 Congruent Polygons S Remember from previous lessons that is something is congruent, that it has the same size and same shape. S Another way to look at it is that you could take any polygon and slide it on top of the same polygon.
Naming Polygons Here we see a triangle. When naming a polygon, start at any point on the figure, then go clockwise or counterclockwise. Using this rule, we can name this triangle six different ways. ABC, ACB, BAC, BCA, CAB, and CBA.
In your team huddle, answer the question below What are all the possible names for the quadrilateral shown at left?
Corresponding Sides and Angles S S S Two polygons can be corresponding if they have the same number of sides and we can pair their parts. Here we have quadrilaterals ADCB and EHGF. How can we prove that they are corresponding?
S Notice that when we named the triangle in the earlier side, we came up with 6 different ways. S Notice that when we named the quadrilateral in the last side, we came up with 8 different ways. S This lets us see a pattern in that however many sides are in a polygon, we multiply the sides by 2 in order to show the different ways that we can either name the polygon or show correspondence.
S Based on this conception, how many ways could we name a: S Pentagon S Octagon S Decagon S Huddle up in your teams and answer these three questions.
Congruency Statements S Whenever we write a congruency statement about two polygons, we have to write the letters of the vertices in the proper order so that they correspond. S How would we write a congruency statement using these quadrilaterals?
In your team huddle, answer the question below S Write a congruence statement about the trapezoids shown below.
Polygon Congruence Postulate S States that two polygons are congruent if and only if there is a correspondence between their sides and angles such that: S Each pair of corresponding angles is congruent. S Each pair of corresponding sides is congruent. S So if you remember from the previous slide, the number of ways to prove congruency is twice the number of sides of the polygon.
S How can we prove that ΔABC ΔDEF?
4.2 Triangle Congruence S There are many different ways to determine that a triangle and congruent. S Almost all of them involve either angles or sides.
SSS Postulate S SSS (Side-Side-Side) S If the sides of one triangle are congruent to the sides of another triangle, then the two triangles are congruent.
SSS Examples
SAS Postulate S SAS (Side-Angle-Side) S If two sides and the included angle in one triangle are congruent to two sides and the included angle in another triangle, then the two triangles are congruent.
SAS Examples
ASA Postulate S ASA (Angle-Side-Angle) S If two angles and the included side in one triangle are congruent to two angles and the included side in another triangle, then the two triangles are congruent.
ASA Examples
4.3 Analyzing Triangle Congruence S In a team huddle, take 60 seconds and discuss the three ways you know to prove triangles congruent, as well as the ways that you know you cannot prove triangles congruent.
AAA Postulate? S Angle-Angle-Angle? S In looking at the triangles on the next slide, what can we prove? What do we need to prove congruency?
S So as we saw from the previous slide, we proved they were congruent through their angles, but not necessarily their sides. S Thus, AAA is not a way to prove congruency.
AAS Theorem S Angle-Angle-Side S If two angles and a side opposite one of these two angles of a triangle are congruent to the corresponding two angles and side in another triangle, then the two triangles are congruent.
Difference between AAS and ASA S AAS is a theorem, ASA is a postulate. S With AAS, the congruent parts must correspond.
HL Theorem S Hypotenuse Leg S If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of another right triangle, then the two triangles are congruent. S What makes the HL Theorem different from the other postulates and theorems we ve looked at?
4.4 Using Triangle Congruence S In a team huddle, fill in the missing information shown below:
S Go back to the triangles in the previous slide. S How many ways can we prove the triangles congruent S Name them.
S If you can prove that all sides of a triangle are congruent and that all angles of a triangle are congruent, you can use CPCTC (Corresponding Parts of Congruent Triangles are Congruent.) S You only use CPCTC after determining every way that two sets of triangles are congruent. S In order to properly use CPCTC in context, we can use a flowchart proof.
S Given: Two segments, MP and NQ, bisect each other at Point O S Prove: MN PQ
Isosceles Triangle S Isosceles triangles have at least two congruent sides. S Made up of two legs and a base. S The angles opposite the legs are base angles. S The angle opposite the base is the vertex angle.
Isosceles Triangle Theorem S If two sides of a triangle are congruent, then the angles opposite those sides are congruent.
Corollary S Corollaries are additional theorems that can be derived from original theorems. S Two common ones are: The measure of each angle of an equilateral triangle is 60 The bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base.