Geometry Regents Lomac Date 11/20 due 11/23 Using Congruent Triangles to prove Quadrilateral Properties
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1 Geometry Regents Lomac Date 11/20 due 11/23 Using Congruent Triangles to prove Quadrilateral Properties 1 Name Per LO: I can prove statements by first proving that triangles are congruent and then using the corresponding parts to prove other relationships, including relationships in quadrilaterals. DO NOW On the back of this packet (1) transparen cies and dry erase markers Triangles and Quadrilaterals Use a compass and straightedge OR tracing paper/plastic to make quadrilaterals A parallelogram is a 4 sided shape with opposite sides parallel. Why do we get a parallelogram when we rotate any triangle around the midpoint of one of its sides? A rectangle is a 4 sided shape with 4 right angles. Why do we get a rectangle when we rotate a right triangle around the midpoint of its hypotenuse? A rhombus is a 4 sided shape with 4 equal sides. Why do we get a parallelogram if we rotate an isosceles triangle around the midpoint of its base? A square is a 4 sided shape with 4 equal sides and 4 right angles. Why do we get a square when we rotate an isosceles right triangle around the midpoint of its hypotenuse? A kite is a 4 sided shape with 2 pairs of adjacent sides that are congruent. Why do we get a kite when we reflect any triangle across its longest side? A trapezoid is a 4 sided shape with at least one pair of parallel opposite sides. Why can t the trapezoid at left be made by rotating or reflecting a triangle?
2 2 (2) Quadrilateral Proofs (a) Use the definition of a parallelogram to prove that opposite sides are congruent. (b) Use the definition of a parallelogram and the information you proved in #4a to prove that the diagonals bisect each other.
3 3 (3) Quadrilateral Proofs (a) Use the definition of a rectangle to prove that it is a parallelogram. (b) Use the definition of a rectangle and anything you have proven so far to prove that the diagonals are congruent.
4 4 (4) Quadrilateral Proofs (a) Use the definition of a rhombus to prove that it is a parallelogram. (b) Use the definition of a rhombus to prove that the diagonals are perpendicular.
5 5 (7) Quadrilateral Proofs (a) Use the definition of isosceles trapezoid to prove that its base angles are congruent. (Use 2 altitudes to make a rectangle and 2 right triangles, show the triangles are congruent, use congruent parts.) (b) Use the information from #7 to prove that the diagonals are congruent. (Use congruent parts and overlapping triangles.)
6 6 (6) (7) Exit Ticket ON THE LAST PAGE Homework (1)Use the definition of a kite (a quadrilateral with 2 pairs of consecutive = sides) to prove that diagonal MP bisects NPQ. (2) First, draw it, then make a conclusion I know that... because... MN bisects AND It is given
7 (7) cont. Homework (3) First, draw it, then make a conclusion I know that... because... 7 BOY and TOY are both 90 It is given (4) Of the boxes 1, 2, 3, and 4, at least one of them is correct and at least one is incorrect. For the incorrect boxes, explain what is incorrect.
8 8 (7) cont. Homework (5) Construct equilateral triangle QRS inscribed in the circle below. (Hint: It might help to construct a regular hexagon first. (6) In the diagram for problem number 7, label the center of the circle C and connect C to each vertex of the triangle. (a) What kind of triangle is triangle QCR? (b) What is the measure of angle QCR? (c) What is the measure of angle CQR?
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