1.3: Writing a Proof | Geometry Honors

How it works

A proof connects each claim to a given fact, a definition, or a proved theorem. State the relevant hypotheses before applying a result. A diagram suggests relationships but does not prove unmarked equalities.

AB=BC,\ AB+BC=AC\ \Rightarrow\ AC=2AB

Worked examples

Example 1

Given that $\displaystyle B$ is the midpoint of $\displaystyle AC$ , prove $\displaystyle AC=2AB$ .

The midpoint definition gives $\displaystyle AB=BC$ . Segment addition gives $\displaystyle AC=AB+BC$ . Substitute $\displaystyle BC=AB$ to obtain $\displaystyle AC=2AB$ .

Example 2

Two angles form a linear pair and are congruent. Prove each is right.

Their sum is $\displaystyle 180^\circ$ . Let each measure be $\displaystyle x$ . Then $\displaystyle 2x=180^\circ$ , so $\displaystyle x=90^\circ$ .

Problems

Problem 1.3.1

Given $\displaystyle AB=CD$ and $\displaystyle CD=2EF$ , prove $\displaystyle AB=2EF$ .

Show worked solution

Substitute $\displaystyle 2EF$ for $\displaystyle CD$ in $\displaystyle AB=CD$ . This is the substitution property of equality.

Problem 1.3.2

Given $\displaystyle m\angle A+m\angle B=90^\circ$ and $\displaystyle m\angle A=35^\circ$ , prove $\displaystyle m\angle B=55^\circ$ .

Show worked solution

Substitute $\displaystyle 35^\circ$ and subtract from both sides: $\displaystyle m\angle B=90^\circ-35^\circ=55^\circ$ .

Problem 1.3.3

Why is a segment appearing horizontal insufficient to prove it perpendicular to a segment that appears vertical?

Show worked solution

A sketch can be distorted. Use a right-angle marking, perpendicularity as a given, or slopes whose product is $\displaystyle -1$ when both slopes exist.

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 1.3.1

Problem 1.3.2

Problem 1.3.3

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