Module 9, Lessons 9.1 and 9.2 - Parallelograms Quadrilateral – A 4-sided polygon Parallelogram – A quadrilateral where each pair of opposite sides are parallel

Module 9, Lessons 9.1 and 9.2 - Parallelograms Facts about Parallelograms – Opposite sides of parallelograms are congruent Module 9, Lessons 9.1 and 9.2 - Parallelograms -In order to prove facts about parallelograms (or most other shapes), we must apply what we know about triangles. -We can do this by creating diagonals (lines that connect the opposite vertices of the figure) Statement Reason ABCD is a parallelogram Given DB≅DB The reflexive property of congruence ∠𝐴𝐷𝐵≅ ∠CBD Definition of Alternate Interior Angles ∠𝐶𝐷𝐵≅ ∠ABD Definition of Alternate Interior Angles ∆ADB ≅ ∆CBD ASA Triangle Congruence Theorem AB≅CD and AD≅CB CPCTC

Module 9, Lessons 9.1 and 9.2 - Parallelograms Facts about Parallelograms – Opposite angles of parallelograms are congruent Module 9, Lessons 9.1 and 9.2 - Parallelograms Statement Reason ABCD is a parallelogram Given DB≅DB The reflexive property of congruence ∠𝐴𝐷𝐵≅ ∠CBD Definition of Alternate Interior Angles ∠𝐶𝐷𝐵≅ ∠ABD Definition of Alternate Interior Angles ∆ADB ≅ ∆CBD ASA Triangle Congruence Theorem ∠A ≅∠C CPCTC

Module 9, Lessons 9.1 and 9.2 - Parallelograms Facts about Parallelograms – Converses Module 9, Lessons 9.1 and 9.2 - Parallelograms

Module 9, Lessons 9.3 and 9.4 – Rectangles, Squares, and Rhombuses Facts about Rectangles, Squares, and Rhombuses All angles are right angles All sides are congruent All angles are right angles & All sides are congruent

Module 9, Lessons 9.3 and 9.4 – Rectangles, Squares, and Rhombuses Facts about Rectangles (diagonals) Diagonals are congruent, and bisect each other The intersection of the diagonals creates two pairs of vertical angles Do you notice something about the four triangles created by the diagonals? Opposite triangles are congruent by SAS! You can use this fact to prove things about rectangles. If a pair of triangles are congruent, then the corresponding parts of those triangles are congruent (CPCTC), thus the opposite sides of the rectangle must be congruent.

Module 9, Lessons 9.3 and 9.4 – Rectangles, Squares, and Rhombuses Facts about Squares (diagonals) Diagonals are congruent, and bisect each other The intersection of the diagonals creates four right angles Do you notice something about the four triangles created by the diagonals? All four triangles are congruent by SAS! If all four triangles are congruent, then the corresponding parts of those triangles are congruent (CPCTC), thus the sides of the square must be congruent.

Module 9, Lessons 9.3 and 9.4 – Rectangles, Squares, and Rhombuses Facts about Rhombuses (diagonals) Diagonals bisect each other but are not necessarily congruent The intersection of the diagonals creates four right angles Do you notice something about the four triangles created by the diagonals? All four triangles are congruent by SAS! If all four triangles are congruent, then the corresponding parts of those triangles are congruent (CPCTC), thus the sides of the rhombus must be congruent.

Module 9, Lessons 9.3 and 9.4 – Rectangles, Squares, and Rhombuses Reminder about parallelograms (diagonals) Diagonals bisect each other but are not necessarily congruent The intersection of the diagonals creates 2 pairs of vertical angles Do you notice something about the four triangles created by the diagonals? Opposite triangles are congruent by SAS! If a pair of triangles are congruent, then the corresponding parts of those triangles are congruent (CPCTC), thus the opposite sides of the parallelogram must be congruent.

Practice proving facts about quadrilaterals Practice proving facts about quadrilaterals. Use the word bank to the right and complete the proof.

Page 427, #’s 6-9 Page 445, #20 Pages 453-454, #’s 7-13 Module 9 Homework Problems Page 427, #’s 6-9 Page 445, #20 Pages 453-454, #’s 7-13 Pages 467, #’s 11 & 12

Module 10 – COORDINATE PROOFS Using DISTANCE on a coordinate plane to prove characteristics about a polygon. Look at the quadrilateral to the right. What type of quadrilateral is it? It looks like it might be a parallelogram, but can we prove it? What do we know about parallelograms? We know that their opposite sides must be congruent! Therefore, if we can show that the opposite sides of the quadrilateral have the same length, then the shape MUST be a parallelogram!

Module 10 – COORDINATE PROOFS Using distance on a coordinate plane to prove characteristics about a polygon. By showing that segment AB ≅ DC and AD ≅ BC, we can prove that this shape is a parallelogram. Use the Pythagorean Theorem or the Distance Formula to find the lengths of each segment: Pythagorean Theorem - 𝒂 𝟐 + 𝒃 𝟐 = 𝒄 𝟐 , where c is the length of the segment you are trying to find. Thus, c = 𝒂 𝟐 + 𝒃 𝟐 The distance formula requires you to determine the values of each coordinate point, whereas the Pythagorean Theorem allows you to simply draw right triangles on the coordinate plane, therefore it is recommended to use the Pythagorean Theorem (demonstrated on the next slide).

Module 10 – COORDINATE PROOFS Pythagorean Theorem - 𝒂 𝟐 + 𝒃 𝟐 = 𝒄 𝟐 , where c is the length of the segment you are trying to find. Thus, c = 𝒂 𝟐 + 𝒃 𝟐 Using the Pythagorean Theorem to show that DC is congruent with AB. Step 1 – create a right triangle with the segment length you are looking for as the hypotenuse Now try the same thing with segment AB to see if it is the same length as DC. Repeat these steps to see if AD and BC are the same length – if so, this is a parallelogram! Step 2 – label the length of the two legs of the right triangle (these will be your ‘a’ and ‘b’ in the Pythagorean Theorem.) 3 1 Step 3 – Plug your values in to the Pythagorean Theorem and solve for ‘c’ which will give you the length of the segment (in this case, segment DC). 3.16 c = 𝟏 𝟐 + 𝟑 𝟐 = 𝟏 + 𝟗 = 𝟏𝟎 ≈ 3.16 NOTE – sometimes you will leave your answer as a square root, but sometimes you will estimate using a calculator. So the segment DC is approximately 3.16 units in length!

Module 10 – COORDINATE PROOFS Using distance on a coordinate plane to prove characteristics about a polygon. This method can be used to determine the nature of a variety of shapes. Simply find the lengths of the sides of any polygon to determine if that polygon is an… Equilateral triangle – all sides must have the same length Isosceles triangle – two sides must have the same length Scalene triangle – all sides must have different lengths Square or Rhombus – all sides must have the same length …..and the list is endless. Try it yourself with the following shapes – determine what shape they are (or aren’t):

Module 10 – COORDINATE PROOFS Using SLOPE on a coordinate plane to prove characteristics about a polygon. Here we have the same quadrilateral from the previous section. We have already shown that this shape is a parallelogram, but is there another way? What ELSE do we know about parallelograms? We know that their opposite sides must be PARALLEL! Therefore, if we can show that the opposite sides of the quadrilateral have the same SLOPE, then the shape MUST be a parallelogram!

Module 10 – COORDINATE PROOFS SLOPE = 𝑹𝒊𝒔𝒆 𝑹𝒖𝒏 = 𝑪𝒉𝒂𝒏𝒈𝒆 𝒊𝒏 𝒚 𝑪𝒉𝒂𝒏𝒈𝒆 𝒊𝒏 𝒙 = 𝒚 𝟐 − 𝒚 𝟏 𝒙 𝟐 − 𝒙 𝟏 There are many methods for calculating slope, but since we already created a right triangle in the previous section, why not do the same thing again?! Step 1 – create a right triangle with the endpoints of the triangle as the vertices of the segment whose slope you are looking for. Now simply create your slope 𝑹𝒊𝒔𝒆 𝑹𝒖𝒏 = 𝟑 𝟏 (or 3) 3 1 The “run” is the length of the horizontal leg, or “1” ONE THING IS MISSING HOWEVER: You must check to see if the slope is POSITIVE or NEGATIVE. If you start with the point farther to the left, and walk to the right, your slope is POSITIVE if you go UP as you go to the right, and NEGATIVE if you go DOWN as you go to the right. The “rise” is the length of the vertical leg, or “3” The slope goes up as you go to the right, so it is positive, thus the slope of DC is 3

Module 10 – COORDINATE PROOFS SLOPE = 𝑹𝒊𝒔𝒆 𝑹𝒖𝒏 = 𝑪𝒉𝒂𝒏𝒈𝒆 𝒊𝒏 𝒚 𝑪𝒉𝒂𝒏𝒈𝒆 𝒊𝒏 𝒙 = 𝒚 𝟐 − 𝒚 𝟏 𝒙 𝟐 − 𝒙 𝟏 Now to see if DC is parallel to AB, calculate the slope of AB the same way. If the slope is the same, the two segments are parallel! Finally, check to see if AD and CB have the same slopes. If so, this is a parallelogram. 3 1 TRY THIS NOW….

Module 10 – COORDINATE PROOFS Using PERPENDICULAR SLOPES on a coordinate plane to discover right angles. Here we have a polygon that appears to be a rectangle. What do we know about rectangles: We know that their opposite sides must be congruent and parallel (just like any parallelogram). But WHAT ELSE do we need to know to be sure this is a rectangle? We need to know that it has four right angles. We already know how to calculate the lengths of the sides of the quadrilateral to show that it is a parallelogram, and we can also calculate slope to show the very same thing, but we need to know a little bit more… we need to see that the slopes of the adjacent sides are PERPENDICULAR!

Module 10 – COORDINATE PROOFS Perpendicular slope…. If two lines are PERPENDICULAR, they intersect at a right angle. Thus, in order for a shape to have right angles, its adjacent sides must be perpendicular to each other. And we can use slope to show that two lines are perpendicular, thus proving that a shape has a right angle! But what is the relationship between perpendicular slopes?!

Module 10 – COORDINATE PROOFS Perpendicular slopes are OPPOSITE INVERSES OF EACH OTHER Let’s look at the slope of the two lines in red. 2 4 3 6 We can draw our right triangles again…. And find the lengths of the legs just like before…. Notice the slope is positive! And calculate the slope of RS = 𝑹𝒊𝒔𝒆 𝑹𝒖𝒏 = 𝟒 𝟐 = 𝟐 𝟏 And the slope of QR = 𝑹𝒊𝒔𝒆 𝑹𝒖𝒏 = 𝟑 𝟔 = 𝟏 𝟐 = - 𝟏 𝟐 𝟐 𝟏 and - 𝟏 𝟐 are opposite inverses of each other, so these two segments are perpendicular to each other! And notice this slope is negative!

Module 10 – COORDINATE PROOFS Using PERPENDICULAR SLOPES on a coordinate plane to discover right angles. Find the slopes of the remaining two sides to see that the adjacent sides are perpendicular to each other, and the opposite sides are parallel.

Module 10 Homework Problems Page 501-502, #’s 2-5 Page 516, #’s 2-5 Page 534, #’s 12-14 Page 544-545, #’s 7-16