1. 12 Chapter Congruence, and Similarity with Constructions
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2. 12-1 Congruence Through Constructions
Geometric Constructions
Constructing Segments
Triangle Congruence
Side, Side, Side Congruence Condition (SSS)
Constructing a Triangle Given Three Sides
Constructing Congruent Angles
Side, Angle, Side Property (SAS)
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3. 12-1 Congruence Through Constructions (continued)
Constructions Involving Two Sides and an Included Angle of a Triangle
Selected Triangle Properties
Construction of the Perpendicular Bisector of a Segment
Construction of a Circle Circumscribed About a Triangle
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4. Congruence and Similarity
Congruent fish and congruent birds have the same shape and same size.
Similar triangles have the same shape but not the same size.
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5. Congruence and Similarity
Similar objects () have the same shape but not necessarily the same size.
Congruent objects () have the same shape and the same size.
Congruent objects are similar, but similar objects are not necessarily congruent.
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Definition
Congruent Segments and Angles
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7. Euclidean Constructions
Given two points, a unique straight line can be drawn containing the points as well as a unique segment connecting the points. (This is accomplished by aligning the straightedge across the points.)
It is possible to extend any part of a line.
A circle can be drawn given its center and radius.
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8. Euclidean Constructions
Any number of points can be chosen on a given line, segment, or circle.
Points of intersection of two lines, two circles, or a line and a circle can be used to construct segments, lines, or circles.
No other instruments (such as a marked ruler, triangle, or protractor) or procedures can be used to perform Euclidean constructions.
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9. Geometric Constructions
Constructing a circle given its radius
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10. Constructing Segments
There are many ways to draw a segment congruent to another segment – using a ruler, tracing the segment, and the method shown below using a straightedge and compass.
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Triangle Congruence
Two figures are congruent if it is possible to fit one figure onto the other so that matching parts coincide.
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Definition
Congruent Triangles
Corresponding parts of congruent triangles are congruent, abbreviated CPCTC.
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Example 12-1
Write an appropriate symbolic congruence for each of the pairs.
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14. Side, Side, Side Congruence Condition (SSS)
If the three sides of one triangle are congruent, respectively, to the three sides of a second triangle, then the triangles are congruent.
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Example 12-2
Use SSS to explain why the given triangles are congruent.
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16. Constructing a Triangle Given Three SidesB″ B′ A′ C′ B′ B A C A′ C′
First construct A′C′ congruent to AC.
Locate the vertex B′ by constructing two circles whose radii are congruent to AB and CB.
Where the circles intersect (either point) is B′.
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Triangle Inequality
The sum of the measures of any two sides of a triangle must be greater than the measure of the third side.
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18. Constructing Congruent Angles
With the compass and straightedge alone, it is impossible in general to construct an angle if given only its measure. Instead, a protractor or a geometry drawing utility or some other measuring tool must be used.
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19. Side, Angle, Side Property (SAS)
If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, respectively, then the two triangles are congruent.
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20. Constructions Involving Two Sides and an Included Angle of a Triangle
First draw a ray with endpoint A′, and then construct A′C′ congruent to AC. Then construct Mark B′ on the side of not containing C′ so that Then connect B′ and C′ to complete the triangle.
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21. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
Example 12-3
Use SAS to show that the given pair of triangles are congruent.
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22. Hypotenuse Leg Theorem
If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent.
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23. Selected Triangle Properties
Any point equidistant from the endpoints of a segment is on the perpendicular bisector of the segment.
Any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment.
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24. Selected Triangle Properties
For every isosceles triangle:
The angles opposite the congruent sides are congruent. (Base angles of an isosceles triangle are congruent).
The angle bisector of an angle formed by two congruent sides contains an altitude of the triangle and is the perpendicular bisector of the third side of the triangle.
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25. Selected Triangle Properties
An altitude of a triangle is the perpendicular segment from a vertex of the triangle to the line containing the opposite side of the triangle.
The three altitudes in a triangle intersect in one point.
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26. Construction of the Perpendicular Bisector of a Segment
Construct the perpendicular bisector of a segment by constructing any two points equidistant from the endpoints of the segment.
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27. Construction of a Circle Circumscribed About a Triangle
By constructing the perpendicular bisectors of any two sides of triangle ABC, the point where the bisectors intersect is the circumcenter.
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