Chapter 7: GEOMETRY Section 3 TRIANGLES

California Standards Measurement and Geometry 2.0: Students identify and describe the properties of 2-dimensional figures. Measurement and Geometry 2.3: Identify angles as VERTICAL, ADJACENT, COMPLEMENTARY, or SUPPLEMENTARY and provide descriptions of these terms.

Key Vocabulary TRIANGLE: A polygon formed by three straight segments. SCALENE: Possessing NO sides that are EQUAL in measurement. ISOSCELES: Possessing 2 sides and 2 angles that are EQUAL to each other. EQUILATERAL: Possessing ALL sides EQUAL to each other. ACUTE: A term that refers to the measure of an angle. If an angle is ACUTE, it simply means that the angles possesses a measure GREATER than 0° but LESS than 90°. RIGHT: A term that refers to an angle equaling EXACTLY 90° OBTUSE: A term that refers to the measure of an angle. If an angle is OBTUSE, it simply means that the angles possesses a measure GREATER than 90° but LESS than 180°. CONGRUENT: CONGRUENT is a term that is used to describe EQUALITY. Here, if sides or angles are CONGRUENT, they possess the same measurement.

What is a TRIANGLE? A TRIANGLE is a polygon. A polygon is a closed geometric figure made of line segments. A TRIANGLE is also known as a TRIGON. This means its made up of 3 line segments. TRIANGLES are traditionally named with 3 capital letters. These 3 lettered points are the vertices or endpoints of the line segments. TRIANGLES possess 3 ANGLES and 3 SIDES. The TOTAL Interior Measure of a TRIANGLE is 180°. Basic Formula: Angle 1 + Angle 2 + Angle 3 = 180°

CLASSIFYING TRIANGLES BY SIDES Triangles can be classified by the MEASURE of their SIDES. Using identified sides or measuring tools, you can use the measures of the sides to correctly classify a triangle. Once the side measures are found, TRIANGLES fall into one of the following categories: SCALENE: NOT A SINGLE SIDE MEASURES THE SAME. ISOSCELES: A PAIR of sides is EQUAL to one another. EQUILATERAL: ALL 3 sides are EQUAL to each other.

CLASSIFYING TRIANGLES BY ANGLES Triangles can be classified by the MEASURE of their ANGLES. Using identified angles or measuring tools, you can use the measures of the angles to correctly classify a triangle. Once the angle measures are found, TRIANGLES fall into one of the following categories: ACUTE: The measure of the ANGLES all are UNDER 90°. OBTUSE: The measure of one of the ANGLES is over 90° but under 180°. RIGHT: The measure of one of the ANGLES is EXACTLY 90°.

MEASURES of a TRIANGLE The INTERIOR Measure of a Triangle ALWAYS equals 180°. Any combination of angle measures can be combined. Recall that the 3 angles will ALWAYS result in 180°. The BASIC FORMULA is the following: Angle 1 + Angle 2 + Angle 3 = 180° Examples: 30° + 70° + 80° = 180° (ACUTE Triangle) 45° + 45° + 90° = 180° (RIGHT Triangle) 20°+45°+115° = 180° (OBTUSE Triangle)

SOLVING FOR MISSING ANGLES When you are solving for the missing angle within a Triangle, you will need to work carefully, using ALL known information and the BASIC FORMULA. Simple Algebra will also be used to solve correctly. Example: There is a triangle where Angle 1 is 40° and Angle 2 is 65°. What is the measure of Angle 3? Angle 1 + Angle 2 + Angle 3 = 180° -Basic Formula 40° + 65° + Angle 3 = 180° -Drop in Known Angles 105° + Angle 3 = 180° -Known Angles are combined -105° -105° -Solve Algebraically Angle 3 = 75° -Result is the answer. 40° + 65° + 75° = 180° -Double Check 180° = 180° -Answer is CORRECT

Finding MORE Angles A Triangle exists. Angle 1 measures 85°. Angle 2 measures 25°.Please find the measure for Angle 3. Angle 1+Angle 2+Angle 3=180° -Basic Formula is used 85° + 25°+ Angle 3 = 180° -Drop in the known Angle Measure 110° + Angle 3 = 180° -Solve Algebraically -110° -110° Angle 3 = 70° -Result is your answer A Triangle exists. Angle 1 measures 47°. Angle 2 measures 78°.Please find the measure for Angle 3. Angle 1+Angle 2+Angle 3=180° -Basic Formula is used 47° + 78°+ Angle 3 = 180° -Drop in the known Angle Measure 125° + Angle 3 = 180° -Solve Algebraically -125° -125° Angle 3 = 55° -Result is your answer

QUICK REVIEW Classifying Triangles by SIDE Measures: There are 4 ways an Angle can be classified as: SCALENE: Not a single side is EQUAL. All are DIFFERENT Measures. ISOSCELES: 2 Sides are EQUAL in Measure. EQUILATERAL: ALL 3 sides are EQUAL in Measure. Classifying Triangles by ANGLE Measures: ACUTE: Angles within the Triangle measure UNDER 90°. OBTUSE: One angle within the Triangle is Over 90° but Under 180°. RIGHT: EXACTLY 90°. Interior Measure of a TRIANGLE: Angle 1+Angle 2+Angle 3 = 180°

Check for Understanding Please determine the BEST answer for the following expression. Carry out ALL work and calculations in your NOTES for later reference Please write your answer on your wipe boards and wait for the teacher’s signal. On the count of 3, hold up your wipe boards.

C4U Question #1 Question #1: A Triangle exists where a pair of sides is EQUAL in measure. What type of Triangle is this? Please work out the problem within your notes Write the correct answer on your wipe board. Wait for Teacher’s Signal.

C4U Question #2 Question #2: A Triangle exists where one of the angles is 103° in measure, and the other two angles are 27° and 50°. What type of Triangle is this? Please work out the problem within your notes Write the correct answer on your wipe board. Wait for Teacher’s Signal.

C4U Question #3 Question #3: A Triangle exists where 2 angles are known. Angle 1 measures 53°. Angle 2 measures 72°. Please find the measure for Angle 3. Please work out the problem within your notes Write the correct answer on your wipe board. Wait for Teacher’s Signal.

C4U Question #4 Question #4: A Triangle exists where 2 angles are known. Angle 1 measures 53°. Angle 2 measures 72°. Please find the measure for the Angle 3. Please work out the problem within your notes Write the correct answer on your wipe board. Wait for Teacher’s Signal.

GUIDED PRACTICE Students will work on a worksheet/book work, focusing only on the problems assigned by the teacher. Work carefully, show your problem solving process, and double check all calculations. Use scratch paper to carry out your work. Once you have completed the assigned problems, please raise your pencil. The teacher will then check your work and release you to complete the independent practice.

INDEPENDENT PRACTICE Once you have been signed off and released to complete Independent Practice, please complete the following assignment: