1. Proving Properties of Triangles and Quadrilaterals
Lesson 6.5
Proving Properties of Triangles and Quadrilaterals
Concept: Proofs in the Coordinate Plane
EQs:
-How do we prove the properties of triangles and quadrilaterals in the coordinate plane? (G.GPE.4)
Vocabulary: Square, Rhombus, Rectangle, Parallelogram, Trapezoid, Isosceles Triangle, Right Triangle, Equilateral Triangle
4.2.4: Fitting Linear Functions to Data

2. Types of Quadrilaterals
Types of Triangles
Types of Quadrilaterals
Properties of Triangles
Properties of Quadrilaterals
Complete the Frayer Diagram with your knowledge of shapes and their properties.
6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

3. -Scalene -Isosceles -Equilateral -Right -Square -Rectangle -Rhombus
Types of Triangles
Types of Quadrilaterals
-Scalene
-Isosceles
-Equilateral
-Right
-Square
-Rectangle
-Rhombus
-Trapezoid
Properties of Triangles
Properties of Quadrilaterals
-Three sides and three angles
-Scalene – No sides the same
-Isosceles – Two sides equal length
-Equilateral – All sides equal length
-Right – Right angle
 -Four sides and four angles
 -Square – Equal & Parallel sides and four right angles
-Rectangle – Two sets of equal and parallel lines; four right angles
-Rhombus - Two sets of equal and parallel lines
-Trapezoid – One set of parallel lines
6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

4. Introduction
In this lesson, we will use the distance formula and what we know about parallel and perpendicular lines to prove the properties of triangles and quadrilaterals in the coordinate plane.
6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

5. Key Concepts to Recall:
Formula to find slope: 𝑦 2 − 𝑦 1 𝑥 2 − 𝑥 1
Parallel lines have the SAME slope and NEVER meet
Perpendicular lines have the OPPOSITE RECIPROCAL slopes and meet at a 𝟗𝟎° angle
The Distance Formula:
6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

6. Guided Practice Example 1
A right triangle is defined as a triangle with 2 sides that are perpendicular. Triangle ABC has vertices A (–4, 8), B (–1, 2), and C (7, 6). Determine if this triangle is a right triangle. When disproving a figure, you only need to show one condition is not met.
6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

7. Guided Practice: Example 1, continued
Plot the triangle on a coordinate plane.
6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

8. Guided Practice: Example 1, continued
Calculate the slope of each side using the general slope formula,
𝒎= 𝒚 𝟐 − 𝒚 𝟏 𝒙 𝟐 − 𝒙 𝟏
2 − 8 −1 − −4
= −6 3
Slope of AB =
=−2
6 − − −1
Slope of BC =
= 4 8
= 1 2
6 − − −4
= −2 11
=− 2 11
Slope of AC =
6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

9. Guided Practice: Example 1, continued Observe the slopes of each side.
The slope of is –2 and the slope of BC is .
The slopes of AB and BC are opposite reciprocals of each other which means they are perpendicular.
6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

10. ✔ Guided Practice: Example 1, continued Make connections.
Right triangles have two sides that are perpendicular.
Triangle ABC has two sides that are perpendicular; therefore, it is a right triangle. ✔
6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

11. Guided Practice Example 2
A square is a quadrilateral with two pairs of parallel opposite sides, consecutive sides that are perpendicular, and all sides congruent, meaning they have the same length. Quadrilateral ABCD has vertices A (–1, 2), B (1, 5), C (4, 3), and D (2, 0). Determine if this quadrilateral is a square.
6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

12. Guided Practice: Example 2, continued
Plot the quadrilateral on a coordinate plane.
6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

13. Guided Practice: Example 2, continued
Show the figure has two pairs of parallel opposite sides.
Calculate the slope of each side using the general slope formula,
6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

14. Guided Practice: Example 2, continued
5 − − −1
= 3 2
Slope of AB =
3 − − 1
= −2 3
=− 2 3
Slope of BC =
0 − − 4
= −3 −2
= 3 2
Slope of CD =
0 − − −1
= −2 3
=− 2 3
Slope of AD =
6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

15. Guided Practice: Example 2, continued Observe the slopes of each side.
The side opposite is The slopes of these sides are the same.
This shows that the quadrilateral has two pairs of parallel opposite sides.
6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

16. Guided Practice: Example 2, continued
Sides and are consecutive sides and their slopes are opposite reciprocals.
This is the case for sides BC and CD ; CD and AD; as well as AB and AD.
Thus, the consecutive sides are perpendicular.
6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

17. Guided Practice: Example 2, continued
Show that the quadrilateral has four congruent sides.
Find the length of each side using the distance formula,
6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

18. Guided Practice: Example 2, continued
Thus, the lengths of all four sides are congruent.
6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

19. ✔ Guided Practice: Example 2, continued Make connections.
Recall: A square is a quadrilateral with two pairs of parallel opposite sides, consecutive sides that are perpendicular, and all sides congruent.
Quadrilateral ABCD has two pairs of parallel opposite sides, the consecutive sides are perpendicular, and all the sides are congruent. It is a square. ✔
6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

20. Guided Practice Example 3
Use the distance formula and slope to determine the shape of the figure. A B C D
(−1, 4)
(3, 2)
(0, −4)
(−4, −2)

21. Guided Practice: Example 3, continued
1. First find the slope of each side.
Calculate the slope of each side using the general slope formula,
6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

22. Guided Practice: Example 3, continued
(2)−(4) (3)−(−1)
= −2 4
=− 1 2
Slope of AB:
(−4)−(2) (0)−(3)
= −6 −3
Slope of BC: =2 A B C D
(−1, 4)
(3, 2)
(0, −4)
(−4, −2)
(−2)−(−4) (−4)−(0)
= 2 −4
=− 1 2
Slope of CD:
(2)−(4) (3)−(−1)
= −6 −3
Slope of AD: =2
6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

23. Guided Practice: Example 3, continued
=− 1 2
Slope of AB:
2. Observe the slopes of each side.
The slopes for AB and CD are the same and BC and DA are the same; this makes these parallel lines!
The slopes for AB and CD are the opposite reciprocal of BC and AD. This makes these lines perpendicular, which means they meet at a 𝟗𝟎° angle!
Slope of BC: =2
=− 1 2
Slope of CD:
Slope of AD: =2
6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

24. Guided Practice: Example 3, continued
3. Find and observe the side lengths of the quadrilateral.
Find the length of each side using the distance formula,
6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

25. Guided Practice: Example 3, continued The length from A to B is:
(−1, 4)
(3, 2)
(0, −4)
(−4, −2) 20
≈4.47
6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

26. Guided Practice: Example 3, continued The length from B to C is:
(−1, 4)
(3, 2)
(0, −4)
(−4, −2) 45
≈6.71
6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

27. Guided Practice: Example 3, continued The length from C to D is:B C D
(−1, 4)
(3, 2)
(0, −4)
(−4, −2) 20
≈4.47
6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

28. Guided Practice: Example 3, continued The length from B to C is:
(−1, 4)
(3, 2)
(0, −4)
(−4, −2) 45
≈6.71
6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

29. Guided Practice: Example 3, continued
Distance of AB ≈4.47
Distance of BC ≈6.71
Distance of CD ≈4.47
Distance of AD ≈6.71
4. Observe the distances of each side.
The distances of AB and CD are both 4.47 and the distances of BC and DA are both 6.71.
This shows us that the opposite sides are of equal length.
6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

30. ✔ Guided Practice: Example 3, continued 5. Make connections.
A rectangle and a square are both quadrilaterals with two pairs of parallel opposite sides and consecutive sides that are perpendicular
A rectangle is a quadrilateral with two pairs of opposite sides of equal length.
Quadrilateral ABCD has two pairs of parallel opposite sides, the consecutive sides are perpendicular, and the opposite sides are of equal length.
This shape is a rectangle. ✔
6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

31. Guided Practice
You Try!
What shape is formed by connecting the points (1, 0), (3, 2) and (5, 0)?
Prove your answer using your knowledge of parallel and perpendicular lines and the distance and midpoint formulas.

32. Since it is a three sided figure, we can eliminate the possibility of a quadrilateral.
Label each point and find the slope of each side:
KL = 2 −2 =−1
LM = 0 4 =0
MK = −2 −2 =1
Notice that none of the slopes are the same or opposite reciprocals. This means there are no parallel or perpendicular lines. K
(3, 2) M L
(1, 0)
(5, 0)
6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

33. K M L Find the length of each side: KL ≈2.23 LM =4 MK ≈2.23
Since there were no parallel or perpendicular lines and since two sides are equal, we know this must be an isosceles triangle. K
(3, 2) M L
(1, 0)
(5, 0)
6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

34. Length Distance Parallel Perpendicular Slope Same Side Opposite
Use the words listed below summarize the lesson for today in a few sentences.
Length
Distance
Parallel
Perpendicular
Slope
Same
Side
Opposite
Consecutive
6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance