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Name: ________________________________ Period: _________________ GEOMETRY REVIEW SHEET Polygon: Poly – Many Gon - Angles Triangle: Tri – 3 Angles Scalene: no equal sides Isosceles: 2 equal sides Right Angle: 90° (perpendicular) Equilateral: Equi – equal Lateral – sides 4 90-degree angles; 4 equal sides Only one pairs of parallel sides Regular: equal sides Pentagon Pente – 5 Gon – angles 4 equal sides Two pairs of parallel sides Quadrilateral: Quad – 4 Lateral – sides Hexagon Hex – 6 Gon – angles Page 1 of 13 PERIMETER AND AREA Perimeter – the distance around the exterior of a figure Key: ADD 𝟓+𝟖+𝟔+𝟒=𝟐𝟑 𝒖𝒏𝒊𝒕𝒔 Area – the number of square units needed to fill the space inside a figure Key: MULTIPLY (or count) 𝟑∗𝟓=𝟏𝟓 𝒖𝒏𝒊𝒕𝒔2 Area Formulas: Area = 𝟏/𝟐 𝒃𝒉 Area = 𝒃𝒉 Composite (combined) Shapes: Perimeter – ONLY the exterior (not the inside lines/height) Area – add the areas of each shape 𝑃 = 6 + 4 + 8 + 6 + 10 + 6 𝑃 = 40 𝑢 1 𝐴 𝑇 = (4 + 10) ∗ 5 2 1 𝐴 𝑇 = (14) ∗ 5 = 35 2 𝐴𝑅 = 10 ∗ 6 = 60 𝑇𝐴 = 35 + 60 = 95𝑢2 Page 2 of 13 Area = 𝟏/(𝒃1+𝒃2)𝒉 TRANSFORMATIONS Translation: SLIDE a shape while maintaining size, shape and orientation Translate the figure 3 units up, 5 units to the right. Reflection: FLIP (mirror) a shape across a line while maintaining the shape and size. Reflect △ 𝐴𝐵𝐶 across the y-axis. Rotation: TURN a shape around a fixed point while maintaining the shape and size. Rotate △ 𝐴𝐵𝐶 around the origin. Dilation: EXPAND (or SHRINK) a shape by a proportional ratio (zoom factor). Dilate △ 𝐴𝐵𝐶 by a zoom factor of 2. Page 3 of 13 ANGLE RELATIONSHIPS Complementary Angles – Two angle measures that add up to 90°. Supplementary Angles – Two angle measures that add up to 180°. Vertical Angles – The two opposite (non-adjacent) angles formed by two intersecting lines; congruent Markings: Equal/congruent angles – Equal/congruent sides – Page 4 of 13 Parallel Lines and Special Angle Relationships: Corresponding Angles: Angles which lie in the same position but at different points of intersection of the transversal. When lines are parallel, they are congruent. Name the other corresponding angles in the figure. Alternate Interior Angles: Angles which lie one on the left and one on the right side of the transversal, and both are between the pair of lines. When lines are parallel, they are congruent. Name the other alternate interior angles in the figure. Same-side Interior Angles: Angles which lie on the same side of the transversal and both are between the pair of lines. When lines are parallel, they are supplementary (= 180°) Name the other same-side interior angles in the figure. Page 5 of 13 TRIANGLE INEQUALITY THEOREM Not all sets of 3 segments can form a triangle! There are limitations… Any side must be less than the sum of the other two sides Any side must be greater than the difference of the other two sides Which of the following sets of 3 lengths cannot form a triangle? 4, 5, 6 3, 3, 3 2, 3, 5 7, 7, 15 8, 9, 16 12, 13, 25 ___________________________________________________________________ PYTHAGOREAN THEOREM Definition – In a right triangle, the sum of the squares of the legs is equal to the square of the hypotenuse. Legs – form the right angle Hypotenuse – longest side (opposite the right angle) Solve for the missing side in the triangle: 𝒂𝟐 + 𝒃𝟐 = 𝒄𝟐 𝟔𝟐 + 𝒃𝟐 = 𝟏𝟐𝟐 𝟑𝟔 + 𝒃𝟐 = 𝟏𝟒𝟒 𝒃𝟐 = 𝟏𝟎𝟖 𝒃 = √𝟏𝟎𝟖 𝒃 = √𝟑𝟔 ∗ 𝟑 𝒃 = 𝟔√𝟑 Page 6 of 13 SIMILARITY Definition – two (or more) figures that have the same shape but a different size. Ratio of Similarity – the ratio between any pair of corresponding sides in similar figures (zoom factor). 𝑎:𝑏 or 𝑎/𝑏 or “𝑎 to 𝑏” Finding corresponding parts: Examine any lettering on the figures – the order will tell you Shapes within shapes: Key: Big to big; small to small Page 7 of 13 DETERMINING TRIANGLE SIMILARITY Similarity means… Corresponding angles have equal measure Corresponding sides are proportional Three conjectures to prove similarity: SSS Similarity (SSS ∼): All three corresponding side lengths share a common ratio. AA Similarity (AA ∼): Two pairs of corresponding angles have equal measures. SAS Similarity (SAS ∼): Two pairs of corresponding sides share a common ratio AND the corresponding angles between those sides have equal measure. Page 8 of 13 Are the triangles below similar? If yes, state the conjecture and state the similarity: DETERMINING TRIANGLE CONGRUENCY Congruency means… Two figures have the same shape AND the same size. In other words, the shapes are similar and the side lengths have a common ratio of 1. Five conjectures to prove congruency… SSS (Side-Side-Side) ≅: If all three pairs of corresponding sides have equal lengths, then the triangles are congruent. Page 9 of 13 More congruency conjectures… SAS (Side-Angle-Side) ≅: If two pairs of corresponding sides have equal lengths AND the angles between them (the included angle) are equal, then the triangles are congruent. ASA (Angle-Side-Angle) ≅: If two angles and the side between them in a triangle are congruent to the corresponding angles and side in another triangle, then the triangles are congruent. AAS (Angle-Angle-Side) ≅: If two pairs of corresponding angles AND a pair of corresponding sides that are not between them have equal measures, then the triangles are congruent. HL (Hypotenuse-Leg) ≅: If the hypotenuse and a leg of one right triangle have the same lengths as the hypotenuse and a leg of another right triangle, then the triangles are congruent. Page 10 of 13 TRIGONOMETRIC RATIOS In any right triangle… The sides are proportional to each other in conjunction with the other acute angles. These ratios are defined in three ways… Three trigonometric ratios… Sine (sin) function The ratio of the opposite side from an angle to the hypotenuse Designated by: 𝐬𝐢𝐧 𝜽 = 𝒐𝒑𝒑𝒐𝒔𝒊𝒕𝒆 𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆 Cosine (cos) function The ratio of the adjacent side from an angle to the hypotenuse Designated by: 𝐜𝐨𝐬 𝜽 = 𝒂𝒅𝒋𝒂𝒄𝒆𝒏𝒕 𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆 Tangent (tan) function The ratio of the opposite side from an angle to the adjacent side of that same angle Designated by: 𝐭𝐚𝐧 𝜽 = 𝒐𝒑𝒑𝒐𝒔𝒊𝒕𝒆 𝒂𝒅𝒋𝒂𝒄𝒆𝒏𝒕 Page 11 of 13 Use SOHCAHTOA as a mnemonic to remember… sin 𝜃 = 4/5 cos 𝜃 = 3/5 tan 𝜃 = 4/3 To find missing sides of right triangles with an angle and a single given side… 𝟒 𝒉 𝟒 . 𝟗𝟎𝟔𝟑 = 𝒉 . 𝟗𝟎𝟔𝟑𝒉 = 𝟒 𝟒 𝒉= ≈ 𝟒. 𝟒𝟏 . 𝟗𝟎𝟔𝟑 sin 𝟔𝟓° = To find an angle given the sides of a triangle… 𝟒 𝟓 sin 𝜃 = 𝟎. 𝟖 𝜽 = sin−𝟏 𝟎. 𝟖 𝜽 ≈ 𝟓𝟑. 𝟏𝟑° sin 𝜽 = Page 12 of 13 SPECIAL RIGHT TRIANGLES Two special conditions exist to quickly find the sides of right triangles… 𝟑𝟎°/𝟔𝟎°/𝟗𝟎° Triangle The ratio of the sides is 𝒙: 𝒙√𝟑: 𝟐𝒙 If 𝑥 = 4, the hypotenuse is 8, and the other leg is 4√3. 𝟒𝟓°/𝟒𝟓°/𝟗𝟎° Triangle The ratio of the sides is 𝒙: 𝒙: 𝒙√𝟐 If the hypotenuse is 6√2, both of the legs are 6. 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