Figure 16.4 shows a parallelogram ABCD with diagonals ¯AC and ¯BD that intersect at M and bisect each other.įigure 16.4 Quadrilateral ABCD with diagonals ¯AC and ¯BD that intersect at M and bisect each other. ¯AB and ¯CD are two segments cut by a transversal ¯ADĪh, the last name game of this series! If you have a quadrilateral that has diagonals that bisect each other, your quadrilateral is a parallelogram. ¯BC and ¯AD are two segments cut by a transversal ¯AB The measures of the interior angles of a quadrilateral add up to 360º Quadrilateral ABCD with ∠A ~= ∠C and ∠B ~= ∠D A similar argument shows that ¯AB ‌ ‌ ¯CD. Because your two angles on the same side of the transversal are supplementary, Theorem 10.10 tells you that ¯BC ‌ ‌ ¯AD. Usually the transversal has been ¯AC, but this time you'll use ¯AB. Now, you can look at this quadrilateral in the following light: ¯BC and ¯AD are two segments cut by a transversal ¯AB. Because the measures of the sums of the interior angles of a quadrilateral add up to 360º, you can show m∠A + m∠B = 180º, or that ∠A and ∠B are supplementary angles. Theorem 16.3: If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.I'll state the theorem and use Figure 16.3 to guide you through your proof.įigure 16.3 Quadrilateral ABCD with ∠A ~= ∠C and ∠B ~= ∠D. The third description of the quadrilateral involved both pairs of opposite angles being congruent.

Once again, the sweet taste of victory! You have named that quadrilateral correctly. ¯BC and ¯AD are two segments cut by a transversal ¯AC Quadrilateral ABCD with ¯AB ~= ¯CD and ¯BC ~= ¯AD It's time to write out the details.įigure 16.2 Quadrilateral ABCD with ¯AB ~= ¯CD and ¯BC ~= ¯AD If we show this for both pairs of opposite sides, then we have a parallelogram by definition. Use the SSS Postulate to show that the two triangles are congruent, and use CPOCTAC to conclude that alternate interior angles are congruent and opposite sides must be parallel. The game plan is to divide the quadrilateral into two triangles using the diagonal ¯AC. We have a parallelogram ABCD with ¯AB ~= ¯CD and ¯BC ~= ¯AD.

Theorem 16.2: If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

Let's write that as a theorem and lay it to rest. In the second “Name That Quadrilateral” game, the quadrilateral had two pairs of congruent sides. Now that you have named that quadrilateral correctly, you can move on to the next quadrilateral. ¯AB and ¯CD are two segments cut by a transversal ¯AC ∠BAC and ∠ACD are alternate interior angles Quadrilateral ABCD with ¯BC ‌ ‌ ¯AD and ¯BC ~= ¯AD. In order to show ΔDAC ~= ΔBCA, you need to use the SAS Postulate. In order to use CPOCTAC, you need to show ΔDAC ~= ΔBCA. The way to show ∠BAC ~= ∠ACD is to use CPOCTAC. If you could show that ∠BAC ~= ∠ACD, then you could conclude that ¯AB ‌ ‌ ¯CD, and you would be done. In this case, ∠BAC and ∠ACD are alternate interior angles. ¯AB and ¯CD are two segments cut by a transversal ¯AC. The second way is to turn it on its side. Then ∠BCA and ∠DAC are alternate interior angles and are congruent because ¯BC ‌ ‌ ¯AD. The first way is to focus on segments ¯BC and ¯AD cut by a transversal ¯AC. You can look at this quadrilateral in two ways. In other words, you need to show that ¯AB ‌ ‌ ¯CD. You need to show that the other pair of opposite sides is parallel. You already know that one pair of opposite sides is parallel. By definition, a parallelogram is a quadrilateral with both pairs of opposite sides parallel.

Theorem 16.1: If one pair of opposite sides of a quadrilateral are parallel and congruent, then the quadrilateral is a parallelogram.

Figure 16.1 will help you visualize the situation.įigure 16.1 Quadrilateral ABCD with ¯BC ‌ ‌ and ¯BC ~= ¯AD. I'll call it a theorem and write a two-column proof. Your first “Name That Quadrilateral” clue involved one pair of opposite sides being parallel and congruent. There are doubters in the car, so you will have to prove it. If you answered “parallelogram” to all of the above, you are correct! Of course, by now you know that it's not enough to claim that I'm thinking of a parallelogram. I'm thinking of a quadrilateral whose diagonals bisect each other. I'm thinking of a quadrilateral with both pairs of opposite angles congruent. I'm thinking of a quadrilateral with both pairs of opposite sides congruent.

I'm thinking of a quadrilateral with one pair of opposite sides parallel and congruent. When Is a Quadrilateral a Parallelogram?.

Quadrilateral abcd is a parallelogram