Worked examples
The question
Question: prove that vertical angles are equal: ∠1 = ∠3.
We must show why ∠1 = ∠3 is always true.
Prove: = ∠1 = ∠3
Careful!
Every row needs a reason: a fact we know is true.
Looking equal in the picture is not enough.
Careful: "it looks equal" is not a reason
A proof is a table.
On the left: what we know.
On the right: why it is true.
1 Use the angle facts
∠1 and ∠2 sit side by side on a straight line.
∠1 + ∠2 = 180°
∠2 and ∠3 are on a straight line too.
∠2 + ∠3 = 180°
2 Reach the goal
Both sums are 180°.
∠1 + ∠2 = ∠2 + ∠3
Take ∠2 away from both sides.
∠1 = ∠3
3 Check
Let's check: every row has a reason.
And the last row is what we had to prove: ∠1 = ∠3.
The proof is done!
Let's look back.
First: use the angle facts.
Then: reach the goal.
Last: check.
▶ Watch it solved
The question
Question: prove that triangles ABC and DEF are congruent.
We know: AB = DE, BC = EF, AC = DF.
We must show why △ABC ≅ △DEF.
Prove: = △ABC ≅ △DEF
Careful!
The letters go in the same order: the first letters match, the second letters match, and so on.
Careful: the order of the letters matters
A with D, B with E, C with F
A proof is a table.
On the left: what we know.
On the right: why it is true.
1 Write what we know
First, what we are told.
AB = DE
Also given: BC = EF
Also given: AC = DF
2 Reach the goal
Three pairs of equal sides: side, side, side.
△ABC ≅ △DEF
3 Check
Let's check: every row has a reason.
And the last row is what we had to prove: △ABC ≅ △DEF.
The proof is done!
Let's look back.
First: write what we know.
Then: reach the goal.
Last: check.
Answer Prove: = △ABC ≅ △DEF
▶ Watch it solved
The question
Question: prove that triangles ABD and CBD are congruent.
We know: AB = CB, AD = CD.
We must show why △ABD ≅ △CBD.
Prove: = △ABD ≅ △CBD
Careful!
The letters go in the same order: the first letters match, the second letters match, and so on.
Careful: the order of the letters matters
A with C, B with B, D with D
A proof is a table.
On the left: what we know.
On the right: why it is true.
1 Write what we know
First, what we are told.
AB = CB
Also given: AD = CD
2 Find more facts
Look at the picture: BD is in both triangles.
BD = BD
3 Reach the goal
Three pairs of equal sides: side, side, side.
△ABD ≅ △CBD
4 Check
Let's check: every row has a reason.
And the last row is what we had to prove: △ABD ≅ △CBD.
The proof is done!
Let's look back.
First: write what we know.
Then: find more facts.
Then: reach the goal.
Last: check.
Answer Prove: = △ABD ≅ △CBD
▶ Watch it solved
