Review on Parallelograms: Opposite angles are congruent, opposite sides are congruent, diagonals bisect each other, and both opposite sides are parallel.
Theorem 5.4- If both pairs of opposite sides in a quadrilateral are congruent, then the quadrilateral is a parallelogram. Here is a proof problem that will prove this.
Given: Segment Ts is congruent to segment QR; segment TQ is congruent to segment SR.
Prove: Quadrilateral QRST is a parallelogram.
Statement Reason
1. 1.Given
2. Segment SQ is congruent to segment SQ 2. Reflexive Prop.
3. Triangle QST is congruent to Triangle SQR 3.SSS postulate
4. Angle 1 is congruent to angle 2; Angle 3 is congruent to angle 4 4.CPCTC
5.Angle 1, angle 2, angle 3, and angle 4 are alternate 5. Def. of alt. int. angles
interior angles
6. Segment TS is parallel to QR; Segment TQ is parallel to segment SR
6. If 2 parallel lines are cut by a transversal and alt. int. angles are congruent, the the lines are parallel.
7. QRST is a parallelogram 7. Def. of parallelogram
Here is another theorem
Theorem 5.5- If one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram.
Given: Segment TS is congruent to segment QR; Segment TS is parallel to segment QR.
Prove: Quadrilateral QRST is a parallelogram.
This one was for homework, so I can't write the answer down.
Theorem 5.6- If both pairs of opposite angles are congruent, then the quadrilateral is a parallelogram.
Given: Angle A is congruent to Angle C: Angle B is congruent to Angle D.
Prove: Quadrilateral ABCD is a parallelogram.
This one was on the homework too, so I'll leave you guys to solve that.
Theorem 5.7- If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
Given: Segment AM is congruent to segment MC; Segment DM is congruent to segment MB.
Prove: ABCD is a parallelogram.
Another one on the homework! Good luck with this one!
So let's review...
The five ways to prove that a quadrilateral is a parallelogram is to show that...
- Both pairs of opposite sides are parallel.
- Both pairs of opposite sides are congruent.
- One pair of opposite sides are congruent and parallel.
- Both pairs of opposite angles are congruent.
- Diagonals bisect each other.
Our homework for today is Page 174 numbers 1-22, 23, and 24, with 25 being optional for a challenge.
Here is a link for more detailed information, as well as extra practice!
http://www.onlinemathlearning.com/parallelogram.html
http://www.sonoma.edu/users/w/wilsonst/courses/math_150/theorems/Parallelograms/default.html
The next scribe is....
Not Sean, Dmitri, J.D., Zahra, Phil K., Phil J., David, Matt, Katie, Krista, Kerryann, Ms. Hunt, Francis, Emma, myself, Hollee, or Olivia.
It's....
Ashley! Good luck on your next scribing!
Have a nice day!
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