Chapter 5 - Test II - 1

angle addition postulate

- Angle Addition Postulate states that if a point S lies in the interior of ∠PQR, then ∠PQS + ∠SQR = ∠PQR.

perpendicular

- Perpendicular means "at right angles". A line meeting another at a right angle, or 90° is said to be perpendicular to it

Always, Sometimes or Never: The circumcenter and the incenter of an equilateral triangle are ... the same point.

- always - Why always - edit here

median of a triangle

- is a segment whose endpoints are a vertex and the midpoint of the opposite side - a triangle's medians are always concurrent

Concurrency of Angle Bisectors Theorem

- the bisectors of the angles of a triangle are concurrent at a point equidistant from the sides of the triangle

postulate (axioms)

Postulate is a true statement, which does not require to be proved.

two lines intersects at a ...

The point at which two or more lines intersect is called the Point of Intersection.

The distance from a vertex to the centroid is ... the length of the median.

- 2/3 - The centroid is the point where the three medians of the triangle intersect. - It has the following properties: The centroid is always located in the interior of the triangle. The centroid is located 2/3 of the distance from the vertex along the segment that connects the vertex to the midpoint of the opposite side.

angle bisector

- A line that splits an angle into two equal angles. ("Bisect" means to divide into two equal parts.)

The mid-segment is ... to the third side and ... its length.

- ALWAYS parallel ... 1/2 - A midsegment of a triangle is a segment connecting the midpoints of two sides of a triangle. - This segment has two special properties. It is always parallel to the third side, and the length of the midsegment is half the length of the third side.

The point of concurrency of the three perpendicular bisectors of a triangle is called the ... and is equidistant from all three ... of the triangle.

- Circumcenter ... vertices - The circumcenter of a triangle is the point where the perpendicular bisectors of the sides intersect. - It is also the center of the circumcircle, the circle that passes through all three vertices of the triangle.

If a point is on an angle bisector, then...

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

If a point is on the perpendicular bisector of a segment, then...

- If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. - Converse of the Perpendicular Bisector Theorem: If a point is equidistant from the endpoints of the segment, then it is on the perpendicular bisector of the segment.

congruent

- The same shape and size. - Two shapes are congruent when you can Turn, Flip and/or Slide one so it fits exactly on the other. - Angles are congruent when they are the same size (in degrees or radians). - Sides are congruent when they are the same length.

perpendicular bisector

- a line that divides a line segment into two equal parts. It also makes a right angle with the line segment. - Each point on the perpendicular bisector is the same distance from each of the endpoints of the original line segment.

equidistant

- a point is equidistant from two objects if it is the same distance from the objects

mid segment of a triangle

- a segment connecting the midpoints of two sides of the triangle

Always, Sometimes or Never: The altitude to the base of an isosceles triangle ... bisects the base.

- always - The Isosceles Triangle Base Theorem states: "The altitude to the base of an isosceles triangle bisects the base."

In a triangle the sum of the lengths of any two sides must ...

- be greater than the measure of the third side. - The Triangle Inequality Theorem states that the sum of any 2 sides of a triangle must be greater than the measure of the third side. - Note: This rule must be satisfied for all 3 conditions of the sides.

In a triangle the longest side is opposite ...

- biggest angle - The longest side is always opposite the largest interior angle. - The shortest side is always opposite the smallest interior angle.

The altitude to the base of an isosceles triangle... the base.

- bisects

incenter or inscribed in

- circle on the inside of the triangle - in the middle where the three lines meet - the center of the circle inscribed in the triangle

circumcenter or circumscribed about

- circle on the outside of the triangle - the center of the circle circumscribed about the triangle

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

Perpendicular Bisector Theorem

- if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment

Triangle Mid-segment Theorem

- if a segment joins the midpoints of two sides of a triangle, then the segment is parallel to the third side and is half as long

Converse of the Triangle Inequality Theorem

- if two angles of a triangle are not congruent (one greater and one lesser), then the side opposite the greater angle is longer than the side opposite the lesser angle

Triangle Inequality Theorem

- if two sides of a triangle are not congruent (one longer and one shorter), then the angle opposite the longer side is greater than the angle opposite the shorter side

The Hinge Theorem (SAS Inequality Theorem)

- if two sides of one triangle are congruent to two sides of another triangle, and the included angles are not congruent, then the longer third side is opposite the larger included angle

Converse of the Hinge Theorem (SSS Inequality Theorem)

- if two sides of one triangle are congruent to two sides of another triangle, and the third sides are not congruent, then the larger included angle is opposite the longer third side

centroid of a triangle

- in a triangle, the point of concurrency of the medians is called the centroid of a triangle - ALWAYS inside the triangle - the point where a triangle of uniform thickness balances - Medians intersect 2/3 the distance from vertex to midpoint of opposite side

The point of concurrency of the three angle bisectors of a triangle is called the ...

- incenter ... sides -The three angle bisectors of a triangle are concurrent in a point equidistant from the sides of a triangle. The point of concurrency of the angle bisectors of a triangle is known as the incenter of a triangle. The incenter will always be located inside the triangle.

Two planes intersect at a ...

- line - Two planes always intersect in a line as long as they are not parallel.

Concurrency of Altitudes Theorem

- lines that contain the altitudes of a triangle are concurrent

A mid-segment is formed by joining ...

- midpoints of two sides - A midsegment of a triangle is a segment connecting the midpoints of two sides of a triangle. - This segment has two special properties. It is always parallel to the third side, and the length of the midsegment is half the length of the third side.

Always, Sometimes or Never: The centroid of a triangle is ... outside the triangle.

- never - Like the centroid, the incenter is always inside the triangle. It is constructed by taking the intersection of the angle bisectors of the three vertices of the triangle.

Always, Sometimes or Never: The incenter of a triangle is ... outside the triangle.

- never - The incenter is the last triangle center we will be investigating. It is the point forming the origin of a circle inscribed inside the triangle. Like the centroid, the incenter is always inside the triangle. It is constructed by taking the intersection of the angle bisectors of the three vertices of the triangle.

Always, Sometimes or Never: The circumcenter of an obtuse triangle is ... inside the triangle.

- never - Why never - edit here

The lines containing the altitudes of a triangle are concurrent at a point called the ...

- orthocenter - The orthocenter is the point where all three altitudes of the triangle intersect. An altitude is a line which passes through a vertex of the triangle and is perpendicular to the opposite side. There are therefore three altitudes in a triangle.

Determining if a angle is an Exterior Angle

- see whether the two sides of the angle are formed by: 1. an extension of a side of the triangle and 2. a side of the triangle

Always, Sometimes or Never: A median in an isosceles triangle is ... a perpendicular bisector.

- sometimes - Why sometimes - edit here

Always, Sometimes or Never: An altitude in a right triangle is ... one of the sides.

- sometimes - Why sometimes - edit here

Always, Sometimes or Never: An altitude of a triangle is ... a median of the triangle.

- sometimes - Why sometimes - edit here - Altitudes can sometimes coincide with a side of the triangle or can sometimes meet an extended base outside the triangle.

Triangle Inequality Theorem for Sum of Lengths of Sides

- sum of the lengths of any sides of a triangle is greater than the length of the third side. - to check whether a triangle can be formed given three lengths, find the sum of pairs of lengths. Each of these sums MUST be greater than the third side.

distance from a point to a line

- the length of the perpendicular segment from the point to the line

orthocenter of a triangle

- the lines that contain the altitudes of a triangle are concurrent at the orthocenter - orthocenter of a triangle can be inside, on, or outside the triangle - form right angles to the sides if a line was drawn through them

Corollary to the Triangle Exterior Angle Theorem

- the measure of an exterior angle of a triangle is greater than the measure of either of its remote interior angles

Concurrency of Medians Theorem

- the medians of a triangle are concurrent at a point that is two-thirds the distance from each vertex to the midpoint of the opposite side

midpoint

- the midpoint is the middle point of a line segment. - it is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment.

Concurrency of perpendicular Bisectors Theorem

- the perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices

altitude of a triangle

- the perpendicular segment from a vertex of the triangle to the line containing the opposite side (or opposite side extended) - and altitude of a triangle can be inside or outside the triangle, or I can be a side of the triangle

point of concurrency

- the point at which the three or more lines intersect is the point of concurrency - to find a point of concurrency, we need to find the point of intersection of only two lines

incenter of a triangle

- the point of where three angle bisectors of the triangle are concurrent - the incenter ALWAYS lies inside the triangle

segment addition postulate

- the segment addition postulate states that given 2 points A and C, a third point B lies on the line segment AC if and only if the distances between the points satisfy the equation AB + BC = AC.

circumcenter of a triangle

- the three perpendicular bisectors of the sides a triangle are concurrent - it CAN lie inside, on, or outside the triangle - on a right triangle the circumcenter is on its hypotenuse - on an acute triangle is inside the triangle - on an obtuse triangle is outside the triangle

Prove that the third length must be greater

- to check whether a triangle can be formed given three lengths, find the sum of pairs of lengths. Each of these sums MUST be greater than the third side.

The distance from a vertex to the centroid is ... the distance from the centroid to the midpoint.

- twice - The centroid of a triangle is the point where the three medians coincide. - The centroid theorem states that the centroid is of the distance from each vertex to the midpoint of the opposite side.

A median is a segment that connects a ... of a triangle with ... The point of concurrency of the three medians is called the ...

- vertex ... midpoint of the opposite side ... centroid - A median of a triangle is a line segment joining a vertex to the midpoint of the opposing side. Every triangle has exactly three medians, one from each vertex, and they all intersect each other at the triangle's centroid.

equidistant from the two lines

- when a point is the same distance from two distinct lines

concurrent

- when three or more lines intersect at one point, they are concurrent

Opposite Ray is ...

A pair of opposite rays are two rays that have the same endpoint and extend in opposite directions.

theorems

A statement that has to be proved.

What are the characteristics of a plane?

It is also called as two-dimensional surface. Any three noncollinear points lie on one and only one plane. A plane has infinite width and length, zero thickness, and zero curvature.

Euler line

The line segment that passes through a triangle's orthocenter, centroid, and circumcenter. These three points are collinear for any triangle.

inequalities in one triangle

The sum of the lengths of two sides of a triangle must always be greater than the length of the third side.