Proving Lines Parallel

Transcription

1 Proving Lines Parallel Proving Triangles ongruent 1

2 Proving Triangles ongruent We know that the opposite sides of a parallelogram are congruent. What about the converse? If we had a quadrilateral whose opposite sides are congruent, then is it also a parallelogram? Step 1: raw a quadrilateral with congruent opposite sides. 2

3 Step 2: raw diagonal. Notice this creates two triangles. What kind of triangles are they? by SSS Step 3: Since the two triangles are congruent, what and? by PT 3

4 Step 4: Now consider to be a transversal. What and? by onverse of lternate Interior ngles Theorem Step 5: y a similar argument, what and? by onverse of lternate Interior ngles Theorem 4

5 If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Property 2 We know that the opposite angles of a parallelogram are congruent. What about the converse? If we had a quadrilateral whose opposite angles are congruent, then is it also a parallelogram? Step 1: raw a quadrilateral with congruent opposite angles. 5

6 Property 2 Step 2: Now assign the congruent angles variables x and y. What is the sum of all the angles? What is the sum of x and y? x y x y x y 360 2x 2y 360 x y 180 x y Property 2 Step 3: onsider to be a transversal. Since x and y are supplementary, what must be true about and? y x x by onverse of onsecutive Interior ngles Theorem y 6

7 Property 2 Step 4: y a similar argument, what and? x y y x by onverse of onsecutive Interior ngles Theorem Property 2 If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. 7

8 Property 3 We know that the diagonals of a parallelogram bisect each other. What about the converse? If we had a quadrilateral whose diagonals bisect each other, then is it also a parallelogram? Step 1: raw a quadrilateral with diagonals that bisect each other. E Property 3 Step 2: What kind of angles are E and E? So what must be true about them? E E E by Vertical ngles ongruence Theorem 8

9 Property 3 Step 3: Now what and? E by SS and PT Property 3 Step 4: y a similar argument, what and? E by SS and PT 9

10 Property 3 Step 5: Finally, if the opposite sides of our quadrilateral are congruent, what our quadrilateral? E is a parallelogram by Property 3 If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. 10

11 Property 4 The last property is not a converse, and it is not obvious. The question is, if we had a quadrilateral with one pair of sides that are congruent and parallel, then is it also a parallelogram? Step 1: raw a quadrilateral with one pair of parallel and congruent sides. Property 4 Step 2: Now draw in diagonal. onsider to be a transversal. What and? by lternate Interior ngles Theorem 11

12 Property 4 Step 3: What must be true about and? What and? by SS and PT Property 4 Step 4: Finally, since the opposite sides of our quadrilateral are congruent, what our quadrilateral? is a parallelogram by 12

13 Property 4 If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram. Example 1 In quadrilateral WXYZ, m W = 42, m X = 138, and m Y = 42. Find m Z. Is WXYZ a parallelogram? Explain your reasoning. 13

14 Example 2 For what value of x is the quadrilateral below a parallelogram? Example 3 etermine whether the following quadrilaterals are parallelograms. 14

15 Example 4 onstruct a flowchart to prove that if a quadrilateral has congruent opposite sides, then it is a parallelogram. Given: Prove: is a parallelogram Summary 15