Properties of Triangles

Questions and Answers

What is the midsegment of a triangle?

The segment connecting the midpoints of two sides of a triangle.

How many midsegments does a triangle have?

Three

What does the three midsegments conjecture state?

The three midsegments of a triangle divide it into four congruent triangles.

What is the triangle midsegment conjecture?

A midsegment of a triangle is parallel to the third side of the triangle and is half the length of the third side. Signup and view all the answers

What is the Triangle Sum Conjecture?

The sum of the measures of the angles in every triangle is 180°. Signup and view all the answers

What does the Third Angle Conjecture state?

If two angles of one triangle are equal in measure to two angles of another triangle, then the third angle in each triangle is congruent. Signup and view all the answers

What is the Isosceles Triangle Conjecture?

If a triangle is isosceles, then the base angles are congruent. Signup and view all the answers

What does the Converse of the Isosceles Triangle Conjecture state?

If a triangle has two congruent angles, then the sides opposite will also be congruent. Signup and view all the answers

What is the Triangle Inequality Conjecture?

The sum of the lengths of any two sides of a triangle is greater than the length of the third side. Signup and view all the answers

How do you find the possible lengths of the third side of a triangle when given the other two lengths?

Take the difference of the two sides < the 3rd side < the sum of the two sides. Signup and view all the answers

What does the Side-Angle Inequality Conjecture state?

In a triangle, if one side is longer than another side, then the angle opposite the longer side is greater than the angle opposite the shorter side. Signup and view all the answers

What is the Triangle Exterior Angle Conjecture?

The measure of an exterior angle of a triangle is equal to the sum of its two remote interior angles. Signup and view all the answers

What is an Exterior Angle?

An angle formed by a side and the extension of an adjacent side. Signup and view all the answers

What is an Adjacent Interior Angle?

Each exterior angle of a triangle has an adjacent interior angle. Signup and view all the answers

What is a Remote Interior Angle?

Non-adjacent angles to an exterior angle; in a triangle there are two remote interior angles for every exterior angle. Signup and view all the answers

What does the Vertex Angle Bisector Conjecture state?

In an isosceles triangle, the bisector of the vertex is also altitude and median. Signup and view all the answers

What is the Equilateral/Equiangular Triangle Conjecture?

Every equilateral triangle is equiangular, and, conversely, every equiangular triangle is equilateral. Signup and view all the answers

What is the Isosceles Triangle Theorem?

If two sides of a triangle are congruent, then the angles opposite are congruent. Signup and view all the answers

What is the Corollary of the Isosceles Triangle Theorem?

The bisector of the vertex angle of an isosceles triangle is perpendicular to the base at its midpoint. Signup and view all the answers

Study Notes

Properties of Triangles

Midsegment: A segment connecting midpoints of two sides, with three midsegments present in a triangle.
Three Midsegments Conjecture: Divides a triangle into four congruent triangles.
Midsegment Conjecture: A midsegment is parallel to the third side and half its length.
Triangle Sum Conjecture: The total measure of angles in a triangle equals 180 degrees.
Third Angle Conjecture: If two angles of one triangle are equal to those of another, the third angles are also congruent.
Isosceles Triangle Conjecture: Base angles of an isosceles triangle are congruent.
Converse of Isosceles Triangle Conjecture: Two congruent angles in a triangle indicate the sides opposite those angles are congruent.
Triangle Inequality Conjecture: The sum of any two side lengths must exceed the length of the third side.
Finding Third Side Lengths: For two known side lengths, the third side's possible length lies between their difference and their sum.
Side-Angle Inequality Conjecture: In a triangle, a longer side corresponds to a larger opposite angle.
Triangle Exterior Angle Conjecture: An exterior angle equals the sum of the two non-adjacent interior angles.
Exterior Angle: Formed by one side and the extension of an adjacent side.
Adjacent Interior Angle: Each exterior angle has a corresponding adjacent interior angle.
Remote Interior Angle: Angles not adjacent to an exterior angle; two remote interior angles exist for each exterior angle.
Vertex Angle Bisector Conjecture: In isosceles triangles, the vertex bisector serves as both altitude and median.
Equilateral/Equiangular Triangle Conjecture: All equilateral triangles are equiangular, and vice versa.
Isosceles Triangle Theorem: Congruent sides lead to congruent angles opposite those sides.
Corollary of the Isosceles Triangle Theorem: The bisector of an isosceles triangle's vertex angle is perpendicular to the base at its midpoint.

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Test your knowledge on the properties and midsegments of triangles with this flashcard quiz. Learn about the definition and significance of midsegments and their conjectures. Perfect for students studying geometry or anyone looking to refresh their knowledge of triangle properties.