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

Monday

What a proof is for
// The parallelogram frame earns its name
⏱ about 20 min

Monday: What a Proof Is For

Comet drops the big wooden frame onto the sawhorses. "The window frame for the flat. I measured it: a parallelogram."

Wren runs the tape along each side. "Opposite sides measure the same, 120 and 90. But measuring is not proving."

"What is the difference?" Comet asks. "The tape says it is a parallelogram."

"The tape says this frame is. A proof says every frame built this way must be." Wren taps the frame. "What do you notice at the corners?"

"Two sticks cross and make four angles. The opposite ones look equal."

Nova hovers over a crossing and projects the four angles in color. "Would you like a hint? Each pair along a stick makes a straight line."

Comet grins. "So the opposite angles have to match. Our designer, help us write that down so it counts."

Comet, Wren and Nova in the Scene Shop with the wooden parallelogram frame on sawhorses, both diagonals stretched with string.

Theorems and proofs

A theorem is a statement that has been proved. A proof is a chain of statements, each with a reason.

The reasons you may use are definitions, postulates (facts agreed without proof), and theorems proved earlier.

A two-column proof lists statements on the left and reasons on the right. A flowing proof tells the same chain as sentences.

Both start from what is given and end with what was to be proved.

  • Given: the facts the problem hands you. Write them first.
  • Definition: what a word means, such as "supplementary angles add to 180°".
  • Postulate: a fact accepted without proof, such as "angles along a straight line add to 180°".
  • Theorem: a fact proved earlier, such as "the angles of a triangle add to 180°" (after Wednesday).

A solved problem to study: vertical angles are congruent

Two crossing lines with a 55 degree angle marked and the angle opposite it marked with a question mark.

Two lines cross at a point and make angles 1, 2 and 3 in a row. Angle 2 sits between the other two.

Angles 1 and 3 are vertical angles: opposite each other at the crossing. Here is the crew's two-column proof that they are congruent.

StatementReason
Angle 1 and angle 2 make a straight line.Given: the two lines cross.
Angle 1 + angle 2 = 180°.Angles along a straight line add to 180°.
Angle 3 + angle 2 = 180°.Angles along a straight line add to 180°.
Angle 1 + angle 2 = angle 3 + angle 2.Both equal 180°.
Angle 1 = angle 3.Subtract angle 2 from both sides.

Why does it work? Both vertical angles share the same neighbor, angle 2, along a straight line.

Each is 180° minus angle 2, so they must be equal. Nothing about the 55° mattered: the proof covers every crossing.

ANGLES AT A CROSSING
  • Read the question.
  • Tap your answer.
Two crossing lines; one angle is 55° and the angle opposite it is marked ?Two sticks of the frame cross. One angle is 55°. What is the angle opposite it, in degrees?
At the same crossing, one angle is 55°. What is the angle next to it along the stick?
Two crossing lines; one angle is 118° and the angle opposite it is marked ?At another crossing one angle is 118°. What is its vertical angle, in degrees?
PUT THE VERTICAL-ANGLE PROOF IN ORDER
  • ?Given: two lines cross, making angles 1, 2 and 3 in a row
  • ?So angle 1 + angle 2 = angle 3 + angle 2
  • ?Subtract angle 2 from both sides: angle 1 = angle 3
  • ?Angle 1 + angle 2 = 180°, because they make a straight line
  • ?Angle 3 + angle 2 = 180°, because they make a straight line too
WHY THIS EXERCISEEvery proof moves from the given to the goal, and each line leans only on the lines above it.
Angles opposite each other where two lines cross are called ____ angles. Type the word.
A statement that has been proved is called a ____. Type the word.
StatementTrue or false?
A postulate is a fact accepted without proof.?
Measuring one frame proves that every frame built that way is a parallelogram.?
Vertical angles are congruent at every crossing, whatever the angle.?
55 + 125 = 180?
A two-column proof lists reasons on the left and statements on the right.?
WHY THIS EXERCISEKnowing what counts as a reason is the first skill of proof.
Draw two crossing lines, label the four angles 1 to 4, and mark the two pairs of vertical angles in two colors.
Try it
Lay two pencils across each other on paper and trace them. Measure one angle, then predict the other three.
Measure to check. Then say out loud why the opposite angles had to match.

Strong start, designer. Tomorrow a brace crosses two parallel rails and eight angles come from one measurement.