Remote Interior Angle Theorem edit
The measure of the exterior angle of a triangle is equal to the sum of the measures of the other two remote interior angles.
Given: In ∆ABC, angle ACD is the exterior angle.
To Prove: m
ACD=m ABC+m BAC
Proof:
| Statements | Reason |
|---|---|
| In ∆ABC, m a+m b+m c=180°---1
|
Sum of the measures of all the angles of a triangle are 180° |
| Also, m b+m d=180°---2
|
Linear Pair Axiom |
| ∴ m a+m c+m b=m b+m d
|
From 1 and 2 |
| ∴ m a+m c+ b b d
|
|
| ∴ m d=m a+m c
| |
| i.e. m ACD=m ABC+m BAC
|
Hence, proved.
Isosceles Triangle Theorem edit
If two sides of a triangle are congruent, then the angles opposite to them are congruent.
Given: In ∆ABC, Side AB
Side AC.
To Prove: ABC ACB.
Construction: Draw the bisector of BAC, intersecting Side BC in point D.
Proof:
| Statements | Reason |
|---|---|
| In ∆ABD & ∆ACD, | |
| Side AB Side AC
|
Given |
| BAD CAD
|
Ray AD is the bisector of BAC
|
| Side AD Side AD
|
Common Side |
| ∴ ∆ABD ∆ACD
|
Side-Angle-Side Test(SAS Test) of Congruency of Triangles |
| ∴ ABD ACD
|
c-a-c-t (Corresponding Angles of Congruent Triangles) |
| i.e ABC ACB
|
B-D-C or same Angle with a different name |
Hence, proved.