splash screen. lesson menu five-minute check (over lesson 6–2) ccss then/now theorems: conditions...
TRANSCRIPT
Five-Minute Check (over Lesson 6–2)
CCSS
Then/Now
Theorems: Conditions for Parallelograms
Proof: Theorem 6.9
Example 1: Identify Parallelograms
Example 2: Real-World Example: Use Parallelograms to Prove Relationships
Example 3: Use Parallelograms and Algebra to Find Values
Concept Summary: Prove that a Quadrilateral Is a Parallelogram
Example 4: Parallelograms and Coordinate Geometry
Example 5: Parallelograms and Coordinate Proofs
Over Lesson 6–2
A.
B.
C.
____?
Over Lesson 6–2
A.
B.
C.
?
Over Lesson 6–2
A. A
B. B
C. C
?
Over Lesson 6–2
An expandable gate is made of parallelograms that have angles that change measure as the gate is adjusted. Which of the following statements is always true?
A. A C and B D
B. A B and C D
C.
D.
Content Standards
G.CO.11 Prove theorems about parallelograms.
G.GPE.4 Use coordinates to prove simple geometric theorems algebraically.
Mathematical Practices
3 Construct viable arguments and critique the reasoning of others.
2 Reason abstractly and quantitatively.
You recognized and applied properties of parallelograms.
• Recognize the conditions that ensure a quadrilateral is a parallelogram.
• Prove that a set of points forms a parallelogram in the coordinate plane.
Identify Parallelograms
Determine whether the quadrilateral is a parallelogram. Justify your answer.
Answer: Each pair of opposite sides has the same measure. Therefore, they are congruent.If both pairs of opposite sides of a quadrilateral are congruent, the quadrilateral is a parallelogram.
A. Both pairs of opp. sides ||.
B. Both pairs of opp. sides .
C. Both pairs of opp. s .
D. One pair of opp. sides both || and .
Which method would prove the quadrilateral is a parallelogram?
Use Parallelograms to Prove Relationships
MECHANICS Scissor lifts, like the platform lift shown, are commonly applied to tools intended to lift heavy items. In the diagram, A C and B D. Explain why the consecutive angles will always be supplementary, regardless of the height of the platform.
Use Parallelograms to Prove Relationships
Answer: Since both pairs of opposite angles of quadrilateral ABCD are congruent, ABCD is a parallelogram by Theorem 6.10. Theorem 6.5 states that consecutive angles of parallelograms are supplementary. Therefore, mA + mB = 180 and mC + mD = 180. By substitution, mA + mD = 180 and mC + mB = 180.
The diagram shows a car jack used to raise a car from the ground. In the diagram, AD BC and AB DC. Based on this information, which statement will be true, regardless of the height of the car jack.
A. A B
B. A C
C. AB BC
D. mA + mC = 180
Use Parallelograms and Algebra to Find Values
Find x and y so that the quadrilateral is a parallelogram.
Opposite sides of a parallelogram are congruent.
Use Parallelograms and Algebra to Find Values
Substitution
Distributive Property
Add 1 to each side.
Subtract 3x from each side.
AB = DC
Use Parallelograms and Algebra to Find Values
Answer: So, when x = 7 and y = 5, quadrilateral ABCD is a parallelogram.
Substitution
Distributive Property
Add 2 to each side.
Subtract 3y from each side.
A. m = 2
B. m = 3
C. m = 6
D. m = 8
Find m so that the quadrilateral is a parallelogram.
Parallelograms and Coordinate Geometry
COORDINATE GEOMETRY Quadrilateral QRST has vertices Q(–1, 3), R(3, 1), S(2, –3), and T(–2, –1). Determine whether the quadrilateral is a parallelogram. Justify your answer by using the Slope Formula.
If the opposite sides of a quadrilateral are parallel, then it is a parallelogram.
Parallelograms and Coordinate Geometry
Answer: Since opposite sides have the same slope, QR║ST and RS║TQ. Therefore, QRST is a parallelogram by definition.
A. yes
B. no
Graph quadrilateral EFGH with vertices E(–2, 2), F(2, 0), G(1, –5), and H(–3, –2). Determine whether the quadrilateral is a parallelogram.
Parallelograms and Coordinate Proofs
Write a coordinate proof for the following statement.
If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
● Begin by placing the vertex A at the origin.
Step 1 Position quadrilateral ABCD on the coordinateplane such that AB DC and AD BC.
● Let AB have a length of a units. Then B hascoordinates (a, 0).
Parallelograms and Coordinate Proofs
● So that the distance from D to C is also a units, letthe x-coordinate of D be b and of C be b + a.
● Since AD BC, position the endpoints of DC so thatthey have the same y-coordinate, c.
Parallelograms and Coordinate Proofs
Step 2 Use your figure to write a proof.
Given: quadrilateral ABCD, AB DC, AD BC
Prove: ABCD is a parallelogram.
Coordinate Proof:
By definition, a quadrilateral is a parallelogram if opposite sides are parallel.
Use the Slope Formula.
Parallelograms and Coordinate Proofs
Answer: So, quadrilateral ABCD is a parallelogrambecause opposite sides are parallel.
Since AB and CD have the same slope and AD and BC have the same slope, AB║CD and AD║BC.
The slope of CD is 0.
The slope of AB is 0.
Which of the following can be used to prove the statement below?If a quadrilateral is a parallelogram, then one pair of opposite sides is both parallel and congruent.
A. AB = a units and DC = a units; slope of AB = 0 and slope of DC = 0
B. AD = c units and BC = c units;
slope of and slope of