Solving geometry problems with reasons

Finding unknown angles in multi-step geometry problems and writing a reason for each step, including using an extra parallel line.

Setting out with reasons

In geometry, the reason is part of the answer. Write each step on its own line with the reason in brackets.

Example 1●●

$AB \parallel CD$. A transversal makes an angle of $65^\circ$ with $AB$. The angle $x^\circ$ is vertically opposite the angle corresponding to $65^\circ$ at $CD$. Find $x$.

The corresponding angle at $CD$ is $65^\circ$.

Corresponding angles, AB ∥ CD

$x = 65$

Vertically opposite angles are equal

Adding a parallel line

Some problems have a "bent" shape between two parallel lines. Draw a third parallel line through the bend, then use alternate angles twice.

Example 2●●●

$AB \parallel CD$. A path goes from line $AB$ to point $E$ and back to line $CD$, making angles of $40^\circ$ with $AB$ and $30^\circ$ with $CD$ (both alternate-style, on the inside). Find $\angle E$.

Draw a line through $E$ parallel to $AB$ and $CD$.

The top part of $\angle E$ is $40^\circ$ and the bottom part is $30^\circ$.

Alternate angles, parallel lines

$\angle E = 40 + 30 = 70^\circ$

Flashcards

What goes with each step?

A reason in brackets.

Bent shape between parallel lines

Draw a third parallel line through the bend.

Reason for co-interior

Co-interior angles, $AB \parallel CD$, are supplementary.

Practice

In Example 2's shape, the angles are $40^\circ$ and $30^\circ$. Find $\angle E$.
Same shape, angles $45^\circ$ and $55^\circ$. Find $\angle E$.
Which is a correct reason?
$AB \parallel CD$. Co-interior angles: one is $50^\circ$. Find the other.
Angles $x^\circ$, $32^\circ$ and a right angle together form a straight angle. Find $x$.
Four equal angles meet at a point with a $180^\circ$ straight angle. Each equal angle is $x^\circ$. Find $x$.
Vertically opposite angles are $(5a - 30)^\circ$ and $(2a + 30)^\circ$. Find $a$.
$AB \parallel CD$. An angle of $115^\circ$ is co-interior with angle $y^\circ$. Find $y$.