Characteristics of Parallelograms Mr. Riddle

Quadrilaterals A quadrilateral is considered to be any polygon with 4 sides. Quadrilaterals Not Quadrilaterals A C E F B G H D

Quadrilaterals Quadrilaterals A polygon with four sides.   Quadrilaterals A polygon with four sides. Angles have a sum of 360°.

Example 1: Finding Angle Measures Find the value of x. 𝑥+72+88+124=360 𝑥+284=360 𝒙=𝟕𝟔° 124° 72° 𝑥° 88°

You try! Find the value of x. 𝑥+47 +90+90+90=360 𝑥+317=360 𝑥=43° (𝑥+47)°

What’s a parallelogram? Knowing that ABCD is a parallelogram, list off ALL of what you believe MIGHT be true about the parallelogram? Ex: Are all the sides the same length? Etc…

Parallelograms Parallelogram   Parallelogram Has all the characteristics of a Quadrilateral Both pairs of Opposite Sides are Parallel Both pairs of Opposite Sides are Congruent

Vocabulary Consecutive Angles – Angles of a polygon that share a side ∠1 and ∠2 are consecutive angles in Parallelogram WXYZ. Can you name two other consecutive angles? 1 2 X W Z Y 3 4

Parallelograms Parallelogram Consecutive angles in a parallelogram are same-side interior angles so…   Parallelogram Has all the characteristics of a Quadrilateral Both pairs of Opposite Sides are Parallel Both pairs of Opposite Sides are Congruent Consecutive Angles are Supplementary …since both pairs of opposite sides are parallel.

Example 2: Using Consecutive Angles Find 𝑚∠𝑆 in 𝑃𝑎𝑟𝑎𝑙𝑙𝑒𝑙𝑜𝑔𝑟𝑎𝑚 𝑅𝑆𝑇𝑊. 𝑚∠𝑆+𝑚∠𝑅=180 𝑚∠𝑆+112=180 𝑚∠𝑆=68° 𝟏𝟏𝟐° S R W T 68°

Parallelograms Parallelogram   Parallelogram Has all the characteristics of a Quadrilateral Both pairs of Opposite Sides are Parallel Both pairs of Opposite Sides are Congruent Consecutive Angles are Supplementary Opposite Angles are Congruent

Example 3: Using Algebra Algebra: Find the value of x in PQRS. Then find QR and PS. 3𝑥−15 R Q S P 2𝑥+3 3𝑥−15=2𝑥+3 𝑥−15=3 𝑥=18 𝑄𝑅=3𝑥−15= 𝟑𝟗 𝑃𝑆 ≅ 𝑄𝑅 so, 𝑃𝑆= 𝟑𝟗

You Try! Find the value of y in Parallelogram EFGH. Then find 𝑚∠𝐸, 𝑚∠𝐺, 𝑚∠𝐹, 𝑎𝑛𝑑 𝑚∠𝐻. 𝒚=𝟏𝟏 𝒎∠𝑬=𝟕𝟎, 𝒎∠𝑮=𝟕𝟎 𝒎∠𝑭=𝟏𝟏𝟎, 𝒎∠𝑯=𝟏𝟏𝟎 (𝟔𝒚+𝟒)° F E H G 𝟑𝒚+𝟑𝟕 °

Parallelograms Parallelogram   Parallelogram Has all the characteristics of a Quadrilateral Both pairs of Opposite Sides are Parallel Both pairs of Opposite Sides are Congruent Consecutive Angles are Supplementary Opposite Angles are Congruent Diagonals bisect each other.

Example 4: Using Algebra Find the values of a and b. 𝒂=𝒃+𝟐 𝒂𝒏𝒅 𝒃+𝟏𝟎=𝟐𝒂−𝟖 So… 𝒃+𝟏𝟎=𝟐 𝒃+𝟐 −𝟖 𝒃𝒚 𝒔𝒖𝒃𝒔𝒕𝒊𝒕𝒖𝒕𝒊𝒐𝒏 𝒃+𝟏𝟎=𝟐𝒃+𝟒−𝟖 𝒃+𝟏𝟎=𝟐𝒃−𝟒 𝟏𝟎=𝒃−𝟒 𝟏𝟒=𝒃 𝒂= 𝟏𝟒 +𝟐 𝒂=𝟏𝟔 𝑎 Y X W Z 𝑏+10 𝑏+2 2𝑎−8

You Try! Solve for x and y. 𝟑𝒚−𝟕=𝟐𝒙 𝒂𝒏𝒅 𝒚=𝒙+𝟏 𝟑 𝒙+𝟏 −𝟕=𝟐𝒙 𝟑𝒙+𝟑−𝟕=𝟐𝒙 𝟑𝒚−𝟕=𝟐𝒙 𝒂𝒏𝒅 𝒚=𝒙+𝟏 𝟑 𝒙+𝟏 −𝟕=𝟐𝒙 𝟑𝒙+𝟑−𝟕=𝟐𝒙 𝟑𝒙−𝟒=𝟐𝒙 𝒙−𝟒=𝟎 𝒙=𝟒 So… 𝒚=𝟒+𝟏 𝒚=𝟓 𝑥+1 Y X W Z 3𝑦−7 𝑦 2𝑥

Proving that a Quadrilateral is a Parallelogram If… Both pairs of opposite sides of a quadrilateral are congruent then the quadrilateral is a parallelogram Both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram One pair of opposite sides of a quadrilateral is both congruent and parallel then the quadrilateral is a parallelogram.

Example 5: Proving Parallelograms – Coordinate Plane B(7,7) C (5,1) D(-6,1) Prove that ABCD is a parallelogram by showing that one pair of opposite sides is both congruent AND parallel.

Example 5: 𝑆𝑙𝑜𝑝𝑒= 𝑦 2 − 𝑦 1 𝑥 2 − 𝑥 1 𝑑= 𝑥 2 − 𝑥 1 2 + 𝑦 2 − 𝑦 1 2 To show that AB is parallel to CD, we find their slopes to see if they’re the same. 𝑆𝑙𝑜𝑝𝑒 𝑜𝑓 𝐴𝐵= 7−7 7− −4 = 0 11 =0 𝑆𝑙𝑜𝑝𝑒 𝑜𝑓 𝐶𝐷= 1−1 5− −6 = 0 11 =0 To show 𝐴𝐵 ≅ 𝐶𝐷 , use distance formula. 𝐴𝐵= −4−7 2 + 7−7 2 = −11 2 + 0 2 𝐴𝐵= 121+0 = 121 =𝟏𝟏 𝐶𝐷= −6−5 2 + 1−1 2 = −11 2 + 0 2 CONCLUSION: Since 𝐴𝐵 ∥ 𝐶𝐷 and 𝐴𝐵 ≅ 𝐶𝐷 , ABCD must be a parallelogram by definition of a parallelogram.

Proving Parallelograms Another way to prove that a quadrilateral is a parallelogram is to use the definition of a parallelogram. Parallelogram: a quadrilateral with both pair of opposite sides parallel. So…we are going to do a proof where we prove that a quadrilateral has two pairs of opposite parallel sides which would then make the quadrilateral a parallelogram. Note: we will use congruent triangles to help us!

Example 6: Proving Parallelograms ∠1≅∠3 𝑋𝑍 ≅ 𝑋𝑍 ∆𝑌𝑍𝑊 𝑆𝐴𝑆≅𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦 1 3 4 𝐶𝑃𝐶𝑇𝐶 5 𝑋𝑌 ∥ 𝑊𝑍 𝐼𝑓 𝑎𝑙𝑡. 𝑖𝑛𝑡. ∠ ′ 𝑠 ≅, 𝑡ℎ𝑒𝑛 𝑙𝑖𝑛𝑒𝑠∥. 6 𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 𝑜𝑓 𝑎 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙𝑜𝑔𝑟𝑎𝑚 2 7

Example 6: Explanation 1.) In order to prove it’s a parallelogram, we must prove BOTH pairs of opposite sides are parallel. 2.) In order to prove that lines are parallel, we need to prove that alternate interior angles are congruent. 3.) In order to prove that alternate interior angles are congruent, we will prove that the triangles are congruent so their corresponding parts (alternate interior angles) are congruent. 4.) In order to prove that triangles are congruent, we’ll use the SAS congruence theorem. So in order… Prove Triangles congruent by SAS Use CPCTC to show alternate interior angles are congruent  Use congruent alternate interior angles to show lines are parallel  Use BOTH sets of parallel sides to prove that WXYZ is a parallelogram. Need more help with proving a quadrilateral is a parallelogram? Go to my website and follow the link under Unit 3b: Proving Parallelograms