warm-up 1.what does it mean for two triangles to be congruent? 2.if a contractor was building a...
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Warm-UpWarm-Up
1. What does it mean for two triangles to be congruent?
2. If a contractor was building a house, how could she or he check to see if all of the roof trusses, which are triangles, were identical?
Warm-UpWarm-Up
1. What does it mean for two triangles to be congruent?
2. If a contractor was building a house, how could she or he check to see if all of the roof trusses, which are triangles, were identical?
4.2 Apply Congruence and Triangles4.2 Apply Congruence and Triangles4.3 Prove Triangles Congruent by SSS4.3 Prove Triangles Congruent by SSS
Objectives:
1.To define congruent triangles
2.To write a congruent statement
3.To prove triangles congruent by SSS
Congruent PolygonsCongruent Polygons
Congruent Triangles Congruent Triangles (CPCTC)(CPCTC)Two triangles are congruent triangles congruent triangles if
and only if the ccorresponding pparts of those ccongruent ttriangles are ccongruent.
• Corresponding sides are congruent
• Corresponding angles are congruent
Congruence StatementCongruence Statement
When naming two congruent triangles, order is very important.
Example 1Example 1
Write a congruence statement for the congruent triangles below.
FAT ~ KDI
Pay Attention to marking
Example 2Example 2
Which polygon is congruent to ABCDE?
ABCDE -?-
QLMNP
Example 3Example 3
Locate points I and S so that BLUE FISH.
HINT: use distance an slope to locate the new points
6
4
2
-2
-4
-5 5
F
H
L
U
E
B
I
S
Properties of Congruent Properties of Congruent TrianglesTriangles
Example 4 Example 4
What is the relationship between <C and <F?
30
30
80
80
E
F
D
A
C
B
They are corresponding and congruent
Third Angle TheoremThird Angle Theorem
If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent.
Example 5Example 5
Now back to the subject of roof trusses. Would it be necessary for the manufacturer of a set of trusses to check that all the corresponding angles were congruent as well as the sides? Answer and explain in your notebook
Example 5Example 5
In other words, is it sufficient that the pieces of wood (the sides of each triangle) are all the same length?
Copying a SegmentCopying a Segment
We’re going to try making two congruent triangles by simply copying the three sides using only a compass and a straightedge. First, let’s learn how to copy a segment.
1. Draw segment AB.
Copying a SegmentCopying a Segment
2. Draw a line with point A’ on one end.
Copying a SegmentCopying a Segment
3. Put point of compass on A and the pencil on B. Make a small arc.
Copying a SegmentCopying a Segment
4. Put point of compass on A’ and use the compass setting from Step 3 to make an arc that intersects the line. This is B’.
Copying a SegmentCopying a Segment
Copying a SegmentCopying a Segment
Click on the button to watch a video of the construction.
Investigation 1Investigation 1
Now apply the construction for copying a segment to copy the three sides of a triangle.
Investigation 1Investigation 1
1. Use your straight edge to construct a triangle.
2. Now draw a line with A’ on one end.
Investigation 1Investigation 1
3. Copy segment AB onto your line to make A’B’.
Investigation 1Investigation 1
4. Put point of compass on A and pencil on C. Copy this distance from A’.
Investigation 1Investigation 1
5. Put point of compass on B and pencil on C. Copy this distance from B’. This is C’
Investigation 1Investigation 1
6. Finish your new triangle by drawing segments A’C’ and B’C’.
Side-Side-Side Congruence Side-Side-Side Congruence PostulatePostulate
SSS Congruence Postulate:SSS Congruence Postulate:If the three sides of one triangle are congruent to
the three sides of another triangle, then the two triangles are congruent.
SSS Congruence PostulateSSS Congruence Postulate
Example 6Example 6
Decide whether the triangles are congruent. Explain your reasoning.
Answer in your notebook
Example 7Example 7
1. AC = AD Given BC = BD2. AB = AB Transitive
Prop.3.. ABC = ABD SSS
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Example 8Example 8
Explain why the bench with the diagonal support is stable, while the one without the support can collapse.