Download Worksheet 12. Other Congruence Properties

Math 128. Construction, Congruence, and Similarity
Other Congruence Properties
T. Judson
Stephen F. Austin State University
Spring 2008
Learning Objectives1
• To understand and be able to apply the angle-side-angle (ASA) property.
• To understand and be able to apply the angle-angle-side (AAS) property.
• To understand and be able to apply the basic properties of geometry to quadrilaterals,
including trapezoids, parallelograms, rectangles, squares, and rhombuses.
Testing Your Knowledge
1. (a) Is it possible to construct two non-congruent triangles with angles measuring 60◦
and 70◦ and an included side of 8 inches?
(b) Is it possible to construct two non-congruent triangles with angles measuring 60◦
and 70◦ and a non-included side of 8 cm?
(c) Is it possible to construct two non-congruent triangles with one acute angle measuring 75◦ and a leg of 5 cm on a side of the 75◦ angle?
(d) Is it possible to construct two non-congruent triangles with angles measuring 30◦ ,
70◦ , and 80◦ ?
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Section 10.2 in R. Billstein, S Libeskind, and J. Lott. A Problem Solving Approach to Mathematics for
Elementary School Teachers, ninth edition. Addison Wesley, Boston, 2007
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2. For each of the following, determine if the given conditions are sufficient to prove that
4P QR ∼
= 4M N O. Justify your answers.
(a)
(b)
(c)
(d)
∠Q ∼
= MN
= ∠N , ∠P ∼
= ∠M , P Q ∼
∼
∼
∼
∠R = ∠O, ∠P = ∠M , QR = N O
∠Q ∼
= ∠N , P R ∼
= M O, P Q ∼
= MN
∠Q ∼
= ∠N , ∠P ∼
= ∠M , ∠R ∼
= ∠O
3. Classify each of the following statements as true or false. If the statement is false, find
a counterexample.
(a) The diagonals of a square a perpendicular bisectors of each other.
(b) If all sides of a quadrilateral are congruent, the quadrilateral is a rhombus.
(c) If a rhombus is a square, then it also
a rectangle.
(d) A square is a trapezoid.
(e) A trapezoid is a parallelogram.
(f) An isosceles trapezoid can also be a
rectangle.
(g) A parallelogram is a trapezoid.
(h) No rectangle is a rhombus.
(i) No trapezoid is a square.
(j) Some squares are trapezoids.
4. (a) Using the definition of a parallelogram and the property that opposite sides of a
parallelogram are congruent, prove that the diagonals of a parallelogram bisect
each other.
(b) Given any line segment drawn through the intersection O of the diagonals of a
parallelogram and intersects the opposite sides of the parallelogram at P and Q,
prove that OP ∼
= OQ.
(c) Prove that SQ ∼
= PT.
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