PROPERTIES OF PLANE FIGURES ANGLES & TRIANGLES

Angles Angles are formed by the intersection of 2 lines, 2 rays, or 2 line segments. The point at which the lines, rays, or line segments intersect is called the vertex. Vertex Vertex Vertex

Measuring Angles The measure of an angle is the amount of rotation in degrees about the vertex from one side to the other. The measure of any angle is between 0º and 180º. The wider the “mouth,” the greater the measure of the angle.

Classifying Angles There are 4 types of angles: acute angle – angle that is greater than 0° and less than 90° right angle – angle that is exactly 90° obtuse angle – angle that is greater than 90° and less than 180° straight angle – angle that is exactly 180°

acute right obtuse straight

Naming Angles Angles are named according to their vertex and the points through which each side passes. Angles can be labeled by only the letter or number that represents their vertex. If more than 1 angle share a vertex, then label the angle with the points on each side of the angle and the vertex. Make sure that the vertex is always the middle letter.  

Homework Complete all of the problems on “PPF Practice Problems 1.” Also, answer the following questions: What do acute people and acute angles have in common? What do obtuse people and obtuse angles have in common?

Angle Addition Postulate When more than 1 angle share a vertex, the sum of the measure of the smaller angles equal the measure of the largest angle. Example 1

Another Example of Applying the Angle Addition Postulate

You Try

Angle Relationships Adjacent Angles – two angles that are next to each other Complementary Angles – two adjacent angles that form a right angle Supplementary Angles – two adjacent angles that form a straight angle Vertical Angles – two angles that are opposite of each other when two lines cross

More Angle Relationships The following relationships are formed when a transversal line intersects two parallel lines: Corresponding Angles – two angles that are on the same side of the transversal line, but one is inside and the other is outside of the parallel lines Alternate Exterior Angles – two angles that are on opposite sides of the transversal line, but both are outside of the parallel lines Alternate Interior Angles – two angles that are on opposite sides of the transversal line, but both are inside of the parallel lines Consecutive (Co-interior) Angles – two angles that are on the same side of the transversal line and are both inside of the parallel lines

Adjacent Angles CAB and BAD are adjacent angles

Complementary Angles

Supplementary Angles

Vertical Angles

Corresponding Angles

Alternate Exterior Angles

Alternate Interior Angles

Consecutive (Co-Interior) Angles

You Try Identify the relationship Identify the relationship

Homework Complete all of the problems on PPF Practice Problems 2.

More Info On Angle Relationships Complementary Angles – add up to 90° Supplementary Angles – add up to 180° Vertical Angles – have the same measure Alternate Exterior Angles – have the same measure Alternate Interior Angles – have the same measure Corresponding Angles – have the same measure Consecutive Angles – add up to 180°

You Try Find the measure of angle b Find the measure of angle b

You Try More Find the measure of angle b Solve for x

Homework Complete all of the problems on PPF Practice Problems 3.

Geometric Notations  

More Geometric Notations  

Congruent Angles and Segments  

Parallel Lines  

Perpendicular Lines  

Homework Complete all of the problems on PPF Practice Problems 4

Basic Facts About Triangles Three-sided polygon Each “corner” is a vertex (vertices for plural) The area of a triangle is ½ · base · height The angles in a triangle add up to 180° Triangles are named by their vertices Triangles are classified by their sides and angles

Classifying Triangles by Angles Acute Triangle – all 3 angles are acute Right Triangle – has 1 right angle Obtuse Triangle – has 1 obtuse angle

Classifying Triangles by Sides Equilateral Triangle – all sides and all angles are congruent Isosceles Triangle – 2 sides are congruent and base angles are congruent Scalene Triangle – no sides are congruent

Classifying Triangles by Angles and Sides Right Scalene Acute Scalene Right Isosceles Obtuse Scalene Acute Isosceles Obtuse Isosceles

You Try: Classify Triangles by Their Angles and Sides 4 1 2 5 6 3

Triangle Angle Sum The measure of the unknown angle is 180° – (81° + 58°) = 41° The measure of the unknown angle is 180° – (90° + 40°) = 50°

Triangle Angle Sum Extended This angle is also 102° due to the property of vertical angles This angle is 180° – (102° + 52°) = 26° The measure of the unknown angle is 180° – 26° = 154° due to the property of supplementary angles.

You Try: Find the Measure of the Unknown Angles 1 2 3

Homework Complete all of the problems on PPF Practice Problems 5 and 6. For extra credit you may complete the “Additional Triangle Angle Sum Practice Problems” sheet.

Triangle Congruence Congruent triangles are exactly the same size (same side lengths and angle measures) Corresponding sides are the congruent sides of congruent triangles Corresponding angles are the congruent angles of congruent triangles

Examples of Congruent Triangles Note: We must name the congruent triangles correctly according to the corresponding angles!

You Try: Write a Statement Indicating the Pairs of Triangle Are Congruent 1 2

Proving Triangle Congruence SSS (side-side-side) – if all 3 pairs of corresponding sides are congruent, then the triangles are congruent SAS (side-angle-side) – if 2 pairs of corresponding sides and the pair of corresponding angles between them are congruent, then the triangles are congruent ASA (angle-side-angle) – if 2 pairs of corresponding angles and the pair of corresponding sides between them are congruent, then the triangles are congruent AAS (angle-angle-side) – if 2 pairs of corresponding angles and the pair of corresponding sides not between them are congruent, then the triangles are congruent

Examples of Proving Triangle Congruence SSS ASA SSS ASA SAS AAS SAS

You Try: State Why the Triangles Are Congruent 1 2

Proving Right Triangle Congruence LA (leg-angle) – if a pair of corresponding legs and a pair of corresponding angles other than the right angles are congruent, then the right triangles are congruent LL (leg-leg) – if 2 pairs of corresponding legs are congruent, then the right triangles are congruent HA (hypotenuse-angle) – if the pair of hypotenuses and a pair of corresponding angles other than the right angles are congruent, then the right triangles are congruent HL (hypotenuse-leg) – if the pair of hypotenuses and a pair of corresponding legs are congruent, then the right triangles are congruent

Examples of Proving Right Triangle Congruence LA HA HL HL LL

You Try: State Why the Right Triangles Are Congruent 1 2 3

Homework Complete all of the problems on PPF Practice Problems 7 For extra credit do Triangle Congruence Additional Practice Problems

Properties of Isosceles Triangles Two sides are congruent Base angles are congruent

Using Properties of Isosceles Triangles to Find Angle Measures This angle is 124° (triangle angle sum) This angle is 28° (base angles of isosceles) This angle is also 28° (vertical) This angle is 124° (supplementary) This angle is 28° because it is congruent to the other base angle and together they must add up to 56° So, by the properties of supplementary angles, angle x is 56°.

You Try: Find the Measure of the Unknown Angle

Homework Complete all of the problems on PPF Practice Problems 8. For extra credit do Properties of Isosceles and Equilateral Triangles Additional Practice Problems.

Triangle Inequality Theorem Any side of a triangle is always shorter than the sum of the other two sides but greater than the difference between the other two sides. The shortest side of a triangle is always greater than the difference between the other two sides, and the longest side of a triangle is always less than the sum of the other two sides.

Example of Using The Triangle Inequality Theorem Example 1 – State whether or not the following three numbers can be lengths of the sides of a triangle: 8, 8, 14 Solution – 1. Check to see if the longest side is less than the sum of the other two sides. 14 < 8 + 8 ? Yes 2. Check to see if the shortest side is greater than the difference between the other two sides. 8 > 14 – 8? Yes Therefore, these three numbers can be lengths of the sides of a triangle.

You Try: State If The Three Numbers Can Be Lengths of the Sides of a Triangle 13, 16, 20 2, 12, 22 4, 11, 15

Finding The Range of Possible Lengths of The Third Side of a Triangle Example 1 – What is the range of possible values for x, the length of the third side of the triangle, when given the lengths of the other two sides? 10, 13, x Solution – 1. x must be less than 13 + 10 = 23 2. x must be greater than 13 – 10 = 3 Therefore, 3 < x < 23.

You Try Find the range of x, the possible lengths of the third side of the triangle, when given the lengths of the other two sides: 9, 13, x Find the range of x, the possible lengths of the third side of the triangle, when given the lengths of the other two sides: 5, 5, x

Homework Complete all of the problems on PPF Practice Problems 9. For extra credit do Triangle Inequality Theorem Additional Practice Problems.

Triangle Sides and Angles   The longer the side, the larger the angle that is across from it. The larger the angle, the longer the side that is across from it.  

Example of Ordering Sides According to Angle Measures    

Example of Ordering Angles According Side Lengths    

You Try In ΔSTU, TU = 8 ¼ SU = 8 4/5 ST = 9 Order the angles from smallest to largest. 1. Order the sides of the triangle from shortest to longest. 2. Order the angles of the triangle from smallest to largest.

Homework Complete all of the problems on PPF Practice Problems 10.

The Pythagorean Theorem Only applies to right triangles Used to find the unknown side length of any right triangle c² = a² + b² a and b are the legs, and c is the hypotenuse (the longest side) The hypotenuse is always across from the right angle. a c b 16² = x² + 10² 16² - 10² = x² 256 – 100 = x² 156 = x² √156 = x 2√39 = x ≈ 12.5

You Try Use the Pythagorean Theorem to find the unknown side length of the following triangles: 1 2

Real-world Application of the Pythagorean Theorem Ex 1: A ladder that is 13 feet long is placed against a wall such that the base of the ladder is 5 feet from the wall. How many feet above the ground is the top of the ladder?

Homework Complete all of the problems on PPF Practice Problems 11.

The Pythagorean Theorem and The Area of Triangle The area of a triangle is ½ · base · height The base and the height of a right triangle are its legs Ex 1: Find the area of the triangle Solution: Step 1 – use the Pythagorean to find the length of the other leg, which is also the height of the triangle Step 2 – plug the values for the base and the height into the formula of the area of triangles.

You Try: Find the Area of the Triangles 1 2

Extension of the Pythagorean Theorem and Area of Triangles Find the area of the triangle.

You Try Find the area of the triangle.

Homework Complete all of the problems on PPF Practice Problems 12. Study for quiz. For extra credit do The Pythagorean Theorem and the Area of a Triangle Additional Practice Problems.

45°-45°-90° Right Triangles Also known as right isosceles triangles Legs are congruent Hypotenuse is √2 times longer than the legs

Finding Missing Side Lengths of 45°-45°-90° Right Triangles Example 1 – Find the missing side lengths. Leave your answer as a radical in simplest form if possible. Solution - Hint – Since we know the length of the hypotenuse, we must find the length of the legs, which are congruent to one another, by dividing the hypotenuse by √2.

Finding Missing Side Lengths of 45°-45°-90° Right Triangles Solution - Example 2 – Find the missing side lengths. Leave your answers as radicals in simplest form if possible. Hint – Since we know the length of one leg, we know the length of the other leg as well because they are congruent. To find the length of the hypotenuse, we must multiply the length of the leg by √2.

Rules To Finding Missing Side Lengths of 45°-45°-90° Right Triangles Rule 1 – If you know the length of the hypotenuse, then divide it by √2 to find the length of the legs. Rule 2 – If you know the length of one of the legs, then multiply it by √2 to find the length of the hypotenuse.

You Try 1. Find the missing side lengths. Leave your answer as a radical in simplest form if possible. 2. Find the missing side lengths. Leave your answer as a radical in simplest form if possible.

Homework Complete all of the problems on PPF Practice Problems 13.

30°-60°-90° Right Triangles The short leg is across from the 30° angle The long leg is across from the 60° angle and is √3 times longer than the short leg The hypotenuse is 2 times longer than the short leg

Finding the Unknown Side Length of 30°-60°-90° Right Triangles The length of the short leg is 5. y is the long leg, so its length is 5√3 x is the hypotenuse, so its length is 5 ∙ 2 = 10 The length of the long leg is √15 y is the short leg, so its length is √15 / √3 = √5 x is the hypotenuse, so its length is 2√5

You Try

Solving Multi-step 30°-60°-90° Right Triangle Problems Example 1 – Find the unknown side length, x. Solution – First find the length of the hypotenuse of the triangle on the right, which is also the long leg of the triangle on the left. 9 ∙ 2 = 18 Next, find the length of x, which is the short leg of the triangle on the left. 18 3 = 18 3 ∙ 3 3 = 18 3 3 = 𝟔 𝟑 18

Solving Multi-step 30°-60°-90° Right Triangle Problems Example 2 – Find the unknown side length, x. Solution – First, find the length of the long leg of the triangle on the left, which is also the hypotenuse of the triangle on the right. 9 3 2 Next, find the short leg of the triangle on the right, which is also x. 9 3 2 ÷ 2 = 9 3 2 ∙ 1 2 = 9 3 4 9 3 2

You Try

Solving Multi-step Special Right Triangle Problems

Solving Multi-step Special Right Triangle Problems