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Geometry 9-12 / Week 04 / Thursday
4/6
Week 04 · Proofs about Lines, Angles, Triangles and Parallelograms

Thursday

Proof Day: the parallelogram frame
// The parallelogram frame earns its name
⏱ about 20 min

Thursday: Proof Day, the Parallelogram Frame

Wren stretches two strings across the frame, corner to corner. "The diagonals. Comet, where do they cross?"

Comet measures from each corner to the crossing. "Each string is split in half. Both of them."

"Coincidence?" Wren asks. "Or a theorem? What does the evidence say?"

"Every frame we built this way did it," Comet says. "But you said measuring is not proving."

Nova projects the frame with one diagonal drawn. Two triangles glow, one above the string, one below.

"Would you like a hint? The frame's sides are parallel, and a diagonal is a transversal."

"Alternate interior angles," Comet says slowly. "Two pairs. And the diagonal is shared. That is ASA!"

Wren nods. "Our designer, write it up. Then the diagonals."

The frame on the sawhorses

A parallelogram is a four-sided figure whose opposite sides are parallel. The crew's frame: AB and CD parallel, BC and DA parallel.

Parallelogram frame ABCD with a 65 degree angle at A and both diagonals crossing at a marked point.

The crew's own tape-measure readings for the frame, in centimeters and degrees. M is where the diagonals cross.

PartReadingPartReading
AB120 cmCD120 cm
BC90 cmDA90 cm
Angle A65°Angle C65°
Angle B115°Angle D115°
Diagonal AC177.8 cmAM and MC88.9 cm each
Diagonal BD115.6 cmBM and MD57.8 cm each

Proof 1: opposite sides and angles are congruent

  1. Given: parallelogram ABCD. Draw diagonal AC.
  2. AB is parallel to CD and AC is a transversal, so angle BAC = angle DCA (alternate interior angles).
  3. BC is parallel to DA and AC is a transversal, so angle BCA = angle DAC (alternate interior angles).
  4. AC is a side of both triangles ABC and CDA, so AC = AC.
  5. Triangle ABC is congruent to triangle CDA by ASA.
  6. So AB = CD, BC = DA and angle B = angle D. Drawing BD the same way gives angle A = angle C.

Proof 2: the diagonals bisect each other

  1. Given: parallelogram ABCD with diagonals AC and BD crossing at M.
  2. AB = CD, by Proof 1.
  3. Angle BAM = angle DCM and angle ABM = angle CDM (alternate interior angles across each diagonal).
  4. Triangle ABM is congruent to triangle CDM by ASA.
  5. So AM = CM and BM = DM. Each diagonal is cut in half at M.

Consecutive angles, like A and B, lie along a transversal between parallel sides, so they add to 180°.

Rectangles are the parallelograms whose diagonals are congruent. In a rectangle every angle is 90°, so triangles ABC and DCB match by SAS. Then AC = BD.

PUT PROOF 1 IN ORDER
  • ?Given: parallelogram ABCD, with diagonal AC drawn
  • ?Angle BAC = angle DCA, alternate interior angles across AB and CD
  • ?Triangle ABC is congruent to triangle CDA by ASA
  • ?So AB = CD and BC = DA
  • ?Angle BCA = angle DAC, alternate interior angles across BC and DA
  • ?AC = AC, the shared side
WHY THIS EXERCISEThe order matters: ASA can only be quoted after both angles and the side between them are in hand.
In Proof 1 the two triangles are congruent by which criterion? Type the three letters.
The angle pairs in Proof 1 are ____ interior angles. Type the word.
A parallelogram's diagonals cut each other in half. The math word is that they ____ each other. Type the word.
USE THE THEOREMS ON THE FRAME
  • Read the question.
  • Tap your answer.
The parallelogram frame ABCD with its diagonals.Angle A of the frame is 65°. What is angle C?
The parallelogram frame ABCD with its diagonals.Angle A of the frame is 65°. What is angle B, next to it?
Diagonal AC of the frame measures 177.8 cm. How long is AM, from corner A to the crossing, in centimeters?
Diagonal BD measures 115.6 cm. How long is MD, in centimeters?
Angle D of the frame is 115°. What is angle A, in degrees? Type the number.
WHY THIS EXERCISEOne corner of a parallelogram tells you all four.

Proved, designer. The frame earned its name. Tomorrow you find these theorems around the house and review the week.

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