Geometry Final Exam Review Notes

Polygon Angle Sum

• Polygon Interior Angle Theorem

  • The sum of the measures of the interior angles of a convex $n$-gon is given by the formula: $(n-2) \cdot 180$, where $n$ is the number of sides.
    *Example: For a pentagon ($n = 5$), the sum of the interior angles is $(5-2) \cdot 180 = 3 \cdot 180 = 540$ degrees.
    *What is the sum of the measures of the interior angles of a pentagon? $180( <strong><em></strong></em> − 2) =$

• Polygon Exterior Angle Theorem

  • The sum of the measures of the exterior angles of any convex polygon, one angle at each vertex, is always 360 degrees.
    What is the sum of the exterior angles of a regular undecagon? = 360 degrees.
    *What is the measure of each exterior angle of a regular 20-gon? 360/ ____ =

• Measure of Each Interior Angle in a Regular Polygon

  • To find the measure of each interior angle in a regular polygon, divide the sum of the interior angles by the number of sides ($n$).
  • Formula: $(\frac{(n-2) \cdot 180}{n})$
    *Example: For a regular octagon ($n = 8$), each interior angle measures $(\frac{(8-2) \cdot 180}{8} = \frac{6 \cdot 180}{8} = \frac{1080}{8} = 135$ degrees.
    *What is the measure of each interior angle in a regular octagon? $180( <strong><em><em></em></strong> −2) <strong><em></em></em></strong> =$

• Measure of Each Exterior Angle in a Regular Polygon

  • To find the measure of each exterior angle in a regular polygon, divide 360 degrees by the number of sides ($n$).
  • Formula: $(\frac{360}{n})$

• Finding the Number of Sides Given an Exterior Angle

  • If one exterior angle of a regular polygon measures 9 degrees, the number of sides can be found using the formula: $n = \frac{360}{\text{exterior angle}}$. Thus, $n = \frac{360}{9} = 40$ sides.
  • If one exterior angle of a regular polygon measures , how many sides does the polygon 9° have? = $360/ 𝑛$

• Finding the Number of Sides Given an Interior Angle

  • If one interior angle of a regular polygon measures 108 degrees, the number of sides can be found using the formula: $\text{interior angle} = \frac{180(n-2)}{n}$. Solving for $n$:
    $108 = \frac{180(n-2)}{n}$
    $108n = 180n - 360$
    $72n = 360$
    $n = 5$
  • Therefore, the polygon has 5 sides (a pentagon).
    *If one interior angle of a regular polygon measures , how many sides does the 108° polygon have? = $180(𝑛−2) /𝑛$

• Finding Unknown Angles in Polygons

  • To find the value of $x$ when given interior angles of a polygon, use the formula for the sum of interior angles: $(n-2) \cdot 180$. Set up an equation where the sum of the given angles plus $x$ equals the calculated sum, then solve for $x$.
    *Find the value of x. (interior angles) $180( <strong><em></strong></em> − 2) = + + + + + = 2$

Triangle Inequality Theorem

• The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

List the sides and angles in order from smallest to largest:

Properties of Parallelograms

Compare Parallelograms:

Congruence

• Triangle Congruence Theorems
  • SSS (Side-Side-Side): If all three sides of one triangle are congruent to the corresponding three sides of another triangle, then the two triangles are congruent.
  • SAS (Side-Angle-Side): If two sides and the included angle of one triangle are congruent to the corresponding two sides and included angle of another triangle, then the two triangles are congruent.
  • ASA (Angle-Side-Angle): If two angles and the included side of one triangle are congruent to the corresponding two angles and included side of another triangle, then the two triangles are congruent.
  • AAS (Angle-Angle-Side): If two angles and a non-included side of one triangle are congruent to the corresponding two angles and non-included side of another triangle, then the two triangles are congruent.
  • Hypotenuse-Leg (HL) Theorem: If the hypotenuse and one leg of a right triangle are congruent to the corresponding hypotenuse and leg of another right triangle, then the two triangles are congruent.
    Given , list all congruent angles and sides. ∆𝐴𝐵𝐸≅∆𝑀𝑁𝑃 Write another valid congruency statement:
• CPCTC (Corresponding Parts of Congruent Triangles are Congruent)
  • If two triangles are proven to be congruent, then all of their corresponding parts (angles and sides) are congruent.
    Given , mark the diagram and complete each of the following statements. ∆𝑆𝑇𝑈≅∆𝐾𝐿𝑀 a. e. _ 𝑇𝑈≅ ∠𝑇≅
    b. f. _ 𝐾𝑀≅ ∠𝑆≅
    c. g. 𝐿𝐾≅ ∆𝑈𝑆𝑇≅
    Angles Sides

Similarity

• Ratios and Proportions
  • A ratio is a comparison of two quantities. A proportion is an equation stating that two ratios are equal.
• Similar Polygons
  • Polygons are similar if their corresponding angles are congruent and the lengths of their corresponding sides are proportional.
    Find the scale factor of Figure A to Figure B.
• Proportions in Triangles
  • Side Splitter Theorem: If a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally.
  • Triangle Angle Bisector Theorem: If a ray bisects an angle of a triangle, then it divides the opposite side into segments proportional to the other two sides of the triangle.
    If , find the value of x ∆𝐾𝐿𝐽~∆𝑉𝑊𝑈
• Similarity in Right Triangles
  • Geometric Mean: In a right triangle, the altitude from the right angle to the hypotenuse creates two smaller triangles that are similar to each other and to the original triangle.
  • Leg Corollary: If the altitude is drawn to the hypotenuse of a right triangle, then each leg is the geometric mean between the hypotenuse and the segment of the hypotenuse that is adjacent to that leg.
  • Altitude Corollary: If the altitude is drawn to the hypotenuse of a right triangle, then the altitude is the geometric mean between the two segments of the hypotenuse.

Determine if the triangles are similar. If yes, state how (by AA~, SSS~, or SAS~) and complete the similarity statement. a. b. Circle One: AA~ SSS~ SAS~ Not ~ ∆𝑃𝑀𝐾~∆ Circle One: AA~ SSS~ SAS~ Not ~ ∆𝐶𝐴𝐵~∆ c. d. Circle One: AA~ SSS~ SAS~ Not ~ ∆𝑅𝑌𝑁~∆ Circle One: AA~ SSS~ SAS~ Not ~ ∆𝑃𝐴𝑆~∆

Find the value of x (angle bisector)

Right Triangles

• The Pythagorean Theorem and Converse
  • Pythagorean Theorem: In a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (legs).
    • Formula: $a^2 + b^2 = c^2$, where $a$ and $b$ are the lengths of the legs, and $c$ is the length of the hypotenuse.
  • Converse of the Pythagorean Theorem: If the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle.
    • If $a^2 + b^2 = c^2$, then the triangle is a right triangle.
    • If a^2 + b^2 > c^2, then the triangle is an acute triangle.
    • If a^2 + b^2 < c^2, then the triangle is an obtuse triangle.
      Find the missing slide length rounded to the nearest 3 decimal places (Pythagorean Theorem). a.
• Special Right Triangles
  • 45° − 45° − 90° Triangle: In a 45-45-90 triangle, the legs are congruent, and the length of the hypotenuse is $\sqrt{2}$ times the length of a leg.
    • If the length of each leg is $a$, then the length of the hypotenuse is $a\sqrt{2}$.
      Practice with triangles. Fill in the blank with the missing length! 45° − 45° − 90°
  • 30° − 60° − 90° Triangle: In a 30-60-90 triangle, the length of the hypotenuse is twice the length of the shorter leg, and the length of the longer leg is $\sqrt{3}$ times the length of the shorter leg.
    • If the length of the shorter leg is $a$, then the length of the hypotenuse is $2a$, and the length of the longer leg is $a\sqrt{3}$.
      Practice with triangles. Fill in the blank with the missing length! 30° − 60° − 90°

Trigonometry

• Trigonometric Ratios
  • Sine (sin): The ratio of the length of the side opposite the angle to the length of the hypotenuse.
    • $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$
  • Cosine (cos): The ratio of the length of the side adjacent to the angle to the length of the hypotenuse.
    • $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$
  • Tangent (tan): The ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.
    • $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$
      Find the missing side length. Round to the nearest tenth. a. b. c.
• Angles of Elevation and Depression
  • Angle of Elevation: The angle formed by a horizontal line and the line of sight to an object above the horizontal line.
  • Angle of Depression: The angle formed by a horizontal line and the line of sight to an object below the horizontal line.
    Find the measure of the indicated angle to the nearest degree. a. b. c.

Circles

• Area of a Circle
  • The area of a circle is given by the formula: $A = \pi r^2$, where $r$ is the radius of the circle.
• Circumference of a Circle
  • The circumference of a circle is given by the formula: $C = 2 \pi r$, where $r$ is the radius of the circle.
    Find the AREA and CIRCUMFERENCE of each circle. a. b.
• Finding the Radius Given the Circumference
  • Given the circumference $C$, the radius $r$ can be found using the formula: $r = \frac{C}{2 \pi}$.
    Find the RADIUS given the CIRCUMFERENCE of the circle is . 26π 𝑐𝑚
• Finding the Diameter Given the Area
  • Given the area $A$, the radius $r$ can be found using the formula: $r = \sqrt{\frac{A}{\pi}}$. The diameter is twice the radius: $d = 2r$.
    Find the DIAMETER given the AREA of the circle is . 90 𝑚 2

Volume

• Volume of a Rectangular Prism
  • The volume of a rectangular prism is given by the formula: $V = lwh$, where $l$ is the length, $w$ is the width, and $h$ is the height.
• Volume of a Pyramid
  • The volume of a pyramid is given by the formula: $V = \frac{1}{3}Bh$, where $B$ is the area of the base and $h$ is the height.
    Find the volume of each figure: a. b.