Chapter 5 Properties of Triangles

Angle Bisector

A bisector starting at a vertex, and then bisecting that angle, and then drawing a segment to the opposite side of the angle.

Perpendicular Bisector

A line, ray, or segment starting at a side, and finding the perpendicular bisector of that side.

Altitude of a Triangle

A perpendicular segment starting at the vertex, then going perpendicular to the opposite side (creating a 90 degree angle).

Equidistant from Two Points

A point that is the same distance from each point.

Median of a Triangle ("center of balance/gravity")

A segment starting at the vertex of a triangle, then bisecting each opposite side.

Perpendicular Bisector

A segment, ray, line, or plane that is perpendicular to a segment at its midpoint.

A/S/N: Th angle bisectors of a scalene triangle _____ intersect at a single point.

Always

A/S/N: The altitude from the vertex angle of an isosceles triangle is _____ the median.

Always

A/S/N: The median to any side of an equilateral triangle is ____ the angle bisector.

Always

A/S/N: The medians of an obtuse triangle will ___ intersect inside the triangle.

Always

A/S/N: The medians will ____ intersect inside an acute triangle.

Always

A/S/N:The angle bisectors of a right triangle ___ intersect inside the triangle.

Always

Theorem 5.11

IF one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle. (In a triangle, the larger angle is opposite the longer side.)

Converse of the Perpendicular Bisector Theorem

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

Converse of the Angle Bisector Theorem

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

Angle Bisector Theorem

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

Perpendicular Bisector Theorem

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

Theorem 5.10

If one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side. (In a triangle, the longer side is opposite the larger angle.)

Hinge Theorem (SAS /= Theorem)

If two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first is longer than the third side of the second.

Converse of the Hinge Theorem (SSS /= Theorem)

If two sides of one triangle are congruent to two sides of another triangle, and the third side of the first is longer than the third side of the second, then the included angle of the first is larger than the included angle of the second.

In an acute triangle, the altitudes intersect ____ the triangle.

Inside

The centroid of a triangle is located _____ the triangle.

Inside

Midsegment of a Triangle

It is a segment that connects the midpoints of two sides of a triangle.

A/S/N: Obtuse triangles medians will _____ intersect on the triangle.

Never

A/S/N: The medians of a triangle ____ intersect outside the triangle.

Never

A/S/N: The perpendicular bisectors of a right triangle ___ intersect inside the triangle.

Never

A/S/N: The perpendicular bisectors of a right triangle will ____ intersect outside the figure.

Never

A/S/N: The perpendicular bisectors of an acute triangle will ____ intersect on the triangle.

Never

The altitudes of an acute triangle _____ intersect outside the triangle.

Never

In a right triangle, the altitudes intersect ____ the triangle.

On

The point of concurrency for perpendicular bisectors of a right triangle is _____ the triangle.

On

A/S/N: In an obtuse angle, the orthocenter is ____ the triangle.

Outside

In an obtuse triangle, the altitudes intersect _____ the triangle.

Outside

A/S/N: The altitude of a triangle is ____ the perpendicular bisector.

Sometimes

A/S/N: The altitudes of a triangle ______ intersect inside the triangle.

Sometimes

A/S/N: The centroid of a triangle is _____ the circumcenter of the triangle.

Sometimes

A/S/N: The median of a is _____ the perpendicular bisector.

Sometimes

A/S/N: The perpendicular bisector _____ has a vertex as an endpoint.

Sometimes

A/S/N: The perpendicular bisectors of an obtuse triangle will ____ intersect on the triangle.

Sometimes

A/S/N:The perpendicular bisector of a triangle is _____ the same segment as the angle bisector.

Sometimes.

Exterior Angle Inequality

The measure of an exterior angle of a triangle is greater than the measure of either of the two nonadjacent interior angles.

Midsegment Theorem

The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long.

Triangle Inequality

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

An angle bisector is the center of an inscribed circle, where all the lines will meet when drawn, and an angle bisector is equidistant from the sides of the triangle.

What are two important facts regarding angle bisectors?

An perpendicular bisector is the center of a circumscribe circle, where all the lines will meet when drawn, and a perpendicular bisector is equidistant from the vertices of the triangle.

What are two important facts regarding perpendicular bisectors?

From vertex to centroid, 2x is the distance from the centroid to the opposite side.

What is the distance from vertex to centroid using the median of a triangle?

Centroid

What is the point of concurrency for a median of a triangle?

Circumcenter

What is the point of concurrency for a perpendicular bisector?

Incenter

What is the point of concurrency for an angle bisector?

Orthocenter

What is the point of concurrency for the altitude of a triangle?

Point of Concurrency

What is the point of intersection of the lines that are drawn?

Acute, obtuse, and right triangles, using median, point of concurrency (centroid) is always in the center of the triangle.

What is the significance of a median?

Using altitudes, a right triangles orthocenter will be on the triangle.

What is the significance of an altitude with a right triangle?

Using altitudes, an acute triangles orthocenter will be inside the triangle.

What is the significance of an altitude with an acute triangle?

Using altitudes, an obtuse triangles orthocenter will be outside the triangle.

What is the significance of an altitude with an obtuse triangle?