Topics
5. Relationships in Triangles
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United States of AmericaFL
Grade 912

5.02 Perpendicular and angle bisectors

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Lesson

Concept summary

When an angle bisector cuts an angle into two congruent angles, we can use the angle bisector theorem and the converse of the angle bisector theorem to solve problems in angles and triangles.

Equidistant

The same distance from two or more objects.

Angle bisector theorem

If a point is on an angle bisector, then it is equidistant from the two legs of the angle.

An angle and a segment. The first endpoint of the segment is on the vertex of the angle, and the second endpoint is in the interior of the angle. The segment divides the angle into two equal angles. From the second vertex of the segment, one segment is drawn to and perpendicular to one of the legs of the angle, and another segment is drawn to and perpendicular to the other leg of the angle.
Converse of angle bisector theorem

If a point is in the interior of an angle and is equidistant from the legs of the angle, then it lies on the angle bisector.

When a perpendicular bisector cuts a line segment at a right angle and into two congruent segments, we can use the perpendicular bisector theorem and the converse of the perpendicular bisector theorem to solve problems in angles and triangles.

Perpendicular bisector theorem

If a point is on the perpendicular bisector of a line segment, then it is equidistant from the end points of the line segment.

A triangle with two congruent legs. A segment is drawn from the apex of the triangle to a point on the base of the triangle, and that is perpendicular to the base. This segment divides the base of the triangle into 2 congruent segments.
Converse of perpendicular bisector theorem

If a point is equidistant from the end points of a line segment, then it is on the perpendicular bisector of that line segment.

Worked examples

Example 1

Construct a perpendicular bisector for the base of the triangle below.

An acute triangle.

Approach

We can construct a perpendicular bisector for a triangle the same way we did for a line segment.

Solution

To construct a perpendicular bisector we can follow the given steps:

  1. Set the compass to a length wider than half of the length of the base of the triangle.

  2. Draw an arc with one vertex at the center.

  3. Draw an arc with the other vertex at the center.

  4. Draw a line through the points where the two arcs intersect.

An acute triangle with the midpoint of the triangle's base shown. A line perpendicular to the base and passing through the midpoint is drawn. Construction lines are drawn.

Example 2

Find the value of x.

A triangle with leg lengths of 15 and 2 x plus 3. A segment is drawn from the apex of the triangle to a point on the base of the triangle, and that is perpendicular to the base. This segment divides the base of the triangle into 2 congruent segments.

Approach

We can see from the diagram that the vertical line is the perpendicular bisector because it cuts the horizontal line in half and is perpendicular to it.

Solution

Since we have a perpendicular bisector, we can apply the perpendicular bisector theorem which tells us that the lengths with measures of 15 and 2x+3 are equal.

\displaystyle 15\displaystyle =\displaystyle 2x+3 Perpendicular bisector theorem
\displaystyle 12\displaystyle =\displaystyle 2x Subtract 3 from both sides
\displaystyle 6\displaystyle =\displaystyle x Divide both sides by 2

Example 3

Determine whether or not the given diagram is valid. Justify your answer.

A triangle with a segment drawn from the apex of the triangle to a point on the base of the triangle, and that is perpendicular to the base. This segment divides the base of the triangle into 2 congruent segments. The base angles of the triangle have measures of 54 degrees, and 56 degrees.

Solution

The given diagram is not valid.

The angle measures not being equal means that the two non-base sides of the triangle are not congruent, using the converse of the base angles theorem. This means that the vertical line cannot be the perpendicular bisector of the base, using the converse of perpendicular bisector theorem. This contradicts the figure, which shows the vertical line segment being a perpendicular bisector of the base.

Outcomes

MA.912.GR.1.1

Prove relationships and theorems about lines and angles. Solve mathematical and real-world problems involving postulates, relationships and theorems of lines and angles.

MA.912.LT.4.8

Construct proofs, including proofs by contradiction.

MA.912.LT.4.10

Judge the validity of arguments and give counterexamples to disprove statements.

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