Geometry
Chapter 3.1 Angles
Objectives:
• Define what an angle is.
• Define the parts of an angle.
Recall our definition for a ray. A ray is a line segment with a definite starting point and extends into infinity in only one direction.
Opposite rays are two rays that are part of the same line and share the same endpoint.
X Z
XY and XZ are opposite rays. The figure formed by Y opposite rays is called a straight angle.
An angle is the figure formed when two rays share a common endpoint.
Two parts are identified on an angle. The common endpoint is called the vertex and the
two rays are called the sides. R
T S
side
side vertex
Geometry
Chapter 3.1 Angles
Objectives:
• Define what an angle is.
• Define the parts of an angle.
We can now give a formal definition for an angle.
An angle can be named many ways.
R
T S
Definition of an Angle: An angle is a figure formed by two noncollinear rays that have a common endpoint.
∠𝑅𝑆𝑇
∠𝑇𝑆𝑅
∠𝑆
∠2 2
Notice that the vertex letter is the middle letter in the first two names.
Be careful of naming angles by the vertex only if two or more angles share the same vertex.
interior
exterior
Bookwork: page 93; problems 9-26
Geometry
Chapter 3.2 Angle Measure
Objectives:
• Learn how to measure and classify angles.
In geometry angles are measured in units called degrees.
Postulate 3-1 Angle Measure Postulate: For every angle, there is a unique positive number between 0 and 180 called the degree measure of the angle.
m∠ABC = n and 0 < n < 180 A protractor can be used to measure and draw angles of a certain measure.
Postulate 3-2 Protractor Postulate: On a plane, given AB and a number r between 0 and 180, there is exactly one ray with endpoint A extending on each side of AB such that the degree measure of the angle is r.
r
A r B
This is stating that for any degree measure r, there are only two rays, one on each side of AB, that can create an angle that measures r.
The symbol for degree is °
Geometry
Chapter 3.2 Angle Measure
Objectives:
• Learn how to measure and classify angles.
Once the measure of an angle is known, it can be classified based on that measure.
Right Angle: A right angle is an angle with a measure of 90 degrees.
Acute Angle: An acute angle is an angle with a measure less than 90 degrees.
Obtuse Angle: An obtuse angle is an angle with a measure greater than 90 degrees.
Drawing Congruent Angles – Hands-On Geometry page 99
Bookwork: page 100; problems 11-28.
Geometry
Magical Midpoints
Objectives:
• The midpoints of a triangle, when connected, create four congruent triangles.
• The outer triangles of a quadrilateral, when connected inside the midpoint quadrilateral, create a congruent quadrilateral.
See Chapter 3 Investigation on page 102.
Geometry
Chapter 3.3 Angle Addition Postulate
Objectives:
• Learn that the sum of the measures of two smaller angles equal the measure of the larger angle.
If we draw an angle ∠RST.
R
T S
Draw interior point X.
X Draw ray SX.
This creates angles 1 and 2.
1 2
If we measured ∠1 and ∠2 and added them together, would their sum equal ∠RST?
Postulate 3-3 Angle Addition Postulate (A-A Postulate): For any angle PQR, if A is in the interior of ∠PQR, then m∠PQA + m∠AQR = m∠PQR.
If m∠1 + 𝑚∠2 = 𝑚∠3, then 𝑚∠1 = 𝑚∠3 − 𝑚∠2, and 𝑚∠2 = 𝑚∠3 − 𝑚∠1.
Geometry
Objectives:
• Learn that the sum of the measures of two smaller angles equal the measure of the larger angle.
Chapter 3.3 Angle Addition Postulate
You have learned that a line segment has a midpoint that bisects the line segment.
Just the same, every angle has a ray that bisects the angle. This ray is called an angle bisector.
Definition of an Angle Bisector: The bisector of an angle is the ray with its endpoint at the vertex of the angle, extending into the interior of the angle. The bisector separates the angle into two angles of equal measure.
???
This simply states that every angle has a bisector that separates the original angle into two equal angles.
Hand-on Geometry – Drawing an angle bisector page 107
Bookwork: page 108; problems 11-23.
Geometry
Objectives:
• Identify adjacent angles and linear pairs of angles.
Chapter 3.4 Adjacent Angles and Linear Pairs of Angles
When you bisect an angle, you create two angles of equal measure. You also create two angles that share a common side. These angles are called adjacent angles.
Definition of Adjacent Angles: Adjacent angles are angles that share a common side and have the same vertex, but have no common interior points in common.
If the noncommon side of two adjacent angles form a straight line, then these angles are called a linear pair.
Definition of Linear Pair: Two angles form a linear pair if and only if they are adjacent and their noncommon sides are opposite rays.
Graphing Calculator Exploration page 112 Bookwork: page 113; problems 8-22
Geometry
Objectives:
• Identify complementary and supplementary angles.
Chapter 3.5 Complementary and Supplementary Angles
Complementary and Supplementary angles are special sets of angles.
Definition of Complementary Angles: Two angles are complementary if and only if the sum of their degree measures is 90.
If two angles are complementary, then each is said to be a complement of the other.
Complementary angles do not have to share the same vertex or a common side.
Meaning, complementary angles do not have to be adjacent.
Definition of Supplementary Angles: Two angles are supplementary if and only if the sum of their degree measures is 180.
If two angles are supplementary, then each is said to be a supplement of the other.
Supplementary angles do not have to share the same vertex or a common side.
Meaning, supplementary angles do not have to be adjacent.
Postulate 3-4 Supplement Postulate: If two angles form a linear pair, then they are supplementary.
Bookwork: page 120; problems 13-31
Geometry
Objectives:
• Identify and use congruent and vertical angles.
Chapter 3.6 Congruent Angles
Recall that congruent segments have the same measure. Congruent angles also have the same measure.
Definition of Congruent Angles: Two angles are congruent if and only if they have the same degree measure.
When indicating two angles are congruent, an arc is used to show which angles are congruent.
1 2
𝑚∠1 = 𝑚∠2;
∠1 ≅ ∠2
When two lines intersect, four angles are formed. There are two pairs of nonadjacent angles. These pairs are called vertical angles.
1
4
2 3 ∠1 𝑎𝑛𝑑 ∠3 are vertical angles
∠2 𝑎𝑛𝑑 ∠4 are vertical angles
Geometry
Objectives:
• Identify and use congruent and vertical angles.
Chapter 3.6 Congruent Angles
Theorem 3-1 Vertical Angle Theorem: Vertical angles are congruent.
1
4
2 3 ∠1 ≅ ∠3
∠2 ≅ ∠4
If two angles are congruent, what do you think is true about their complementary and supplementary angles?
Theorem 3-2: If two angles are congruent, then their complements are also congruent.
Theorem 3-3: If two angles are congruent, then their supplements are also congruent.
Theorem 3-4: If two angles are complementary to the same angle, then they are congruent.
Theorem 3-5: If two angles are supplementary to the same angle, then they are congruent.
Theorem 3-6: If two angles are congruent and supplementary, then each is a right angle.
Theorem 3-7: All right angles are congruent.
Bookwork: page 126; problems 9-23
Geometry
Objectives:
• Identify and use perpendicular lines.
Chapter 3.7 Perpendicular Lines
Lines that intersect at 90 degrees are perpendicular lines.
2 1
4 3
The square symbol where two lines intersect indicate that the lines are perpendicular. Notice that four right angles are formed from the intersection.
Four pairs of adjacent angles are supplementary. These adjacent angles are also linear pairs because they are opposite rays.
Definition of Perpendicular Lines: Perpendicular lines are lines that intersect to form a right angle.
Look at preparing for proof, top of page 129.
Theorem 3-8: If two lines are perpendicular, then they form four right angles.
Geometry
Objectives:
• Identify and use perpendicular lines.
Chapter 3.7 Perpendicular Lines
If we draw a line m. How lines can be drawn that are perpendicular to line m?
If we draw a point T on line m, how many lines can be drawn through point T?
How many of those lines through point T are perpendicular to line m?
Theorem 3-9: If a line m is in a plane and T is a point on m, then there exists exactly one line in that plane that is perpendicular to m at T.
Bookwork: page 132; problems 8-28.
