Geometry Final Exam 2014 DPSA

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RD
3 CARD MARKING
• CONGRUENT FIGURES
• CONGRUENT TRIANGLES
• PROPERTIES OF POLYGONS
• PROPERTIES OF QUADRILATERALS
WHAT IS A CONGRUENT FIGURE?
• CORRESPONDING SIDES AND ANGLES ARE SAME MEASURE
(CONGRUENT).
• CORRESPONDING IS SIMILAR TO MATCHING SIDES AND ANGLES.
• LOOK FOR TIC MARKS AND ANGLE ARCS,
• LOOK FOR ACTUAL SIDE MEASUREMENTS AND ANGLE MEASUREMENTS IF
AVAILABLE.
CONGRUENT TRIANGLES
• SSS: SIDE, SIDE, SIDE
•
IF THREE SIDES OF ONE TRIANGLE ARE CONGRUENT
TO 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 TWO SIDES AND THE
INCLUDED ANGLE OF ANOTHER TRIANGLES THEN THE
TRIANGLES ARE CONGRUENT.
CONGRUENCE IN TRIANGLES CONT.
ASA: ANGLE, SIDE, ANGLE
• TWO ANGLES AND AN INCLUDED SIDE ARE
CONGRUENT IN TWO TRIANGLES.
AAS: ANGLE, ANGLE, SIDE
• TWO ANGLES AND A NON-INCLUDED SIDE ARE
CONGRUENT IN TWO TRIANGLES.
COMMON TRIANGLES
ISOSCELES
• SUM OF ANGLES 180
• TWO SIDES CONGRUENT
• TWO ANGLES ARE CONGRUENT
• BASE ANGLES ARE THE SAME.
EQUILATERAL
• SUM OF ANGLES 180
• ALL ANGLES ARE 60 DEGREES EACH.
• EACH SIDE IS CONGRUENT.
RIGHT TRIANGLE CONGRUENCE
• HYPOTENUSE-LEG THEOREM (HL)
• IF THE HYPOTENUSE AND A LEG OF ONE RIGHT TRIANGLE ARE CONGRUENT TO THE HYPOTENUSE AND LEG OF
ANOTHER RIGHT TRIANGLE, THEN THE TRIANGLES ARE CONGRUENT.
POLYGON PROPERTIES
•
•
•
POLYGON ANGLE-SUM THEOREM
•
POLYGON EXTERIOR ANGLE SUM-THEOREM
•
π‘‡β„Žπ‘’ π‘ π‘’π‘š π‘œπ‘“ π‘‘β„Žπ‘’ π‘šπ‘’π‘Žπ‘ π‘’π‘Ÿπ‘’π‘  π‘œπ‘“ π‘‘β„Žπ‘’ 𝑒π‘₯π‘‘π‘’π‘Ÿπ‘–π‘œπ‘Ÿ π‘Žπ‘›π‘”π‘™π‘’π‘  π‘œπ‘“ π‘Ž π‘π‘œπ‘™π‘¦π‘”π‘œπ‘› = 360 π‘‘π‘’π‘”π‘Ÿπ‘’π‘’π‘ .
INTERIOR ANGLE OF REGULAR POLYGON THEOREM
•
•
𝑛 − 2 180, π‘€β„Žπ‘’π‘Ÿπ‘’ 𝑛 𝑖𝑠 π‘‘β„Žπ‘’ π‘›π‘’π‘šπ‘π‘’π‘Ÿ π‘œπ‘“ 𝑠𝑖𝑑𝑒𝑠.
𝑛−2 180
𝑛
CLASSIFYING POLYGONS
•
•
•
EQUILATERAL
•
ALL SIDES CONGRUENT
EQUIANGULAR
•
ALL ANGLES CONGRUENT
REGULAR
•
ALL SIDES AND ANGLES ARE CONGRUENT.
POLYGON PREFIXES
• TRI
• QUAD
• PENTA
• HEXA
• HEPTA
• OCTA
• NONA
• DECA
• DODECA
•N
QUADRILATERAL PROPERTIES
• 4 sides
• Sum 360
• 4 angles
PARALLELOGRAMS
• OPPOSITE SIDES ARE PARALLEL
• OPPOSITE SIDES ARE CONGRUENT
• OPPOSITE ANGLES ARE CONGRUENT
• CONSECUTIVE ANGLES ARE SUPPLEMENTARY
• DIAGONALS BISECT EACH OTHER
RECTANGLES
• HAVE ALL THE PROPERTIES OF A PARALLELOGRAM
• EACH ANGLE IS 90 DEGREES
• DIAGONALS ARE EQUAL IN LENGTH
RHOMBI
• ALL THE PROPERTIES OF PARALLELOGRAMS
• FOUR CONGRUENT SIDES
• DIAGONALS ARE PERPENDICULAR
• DIAGONALS BISECT EACH OTHER
• DIAGONALS BISECT EACH ANGLE
SQUARES
• HAVE ALL THE PROPERTIES OF A PARALLELOGRAM, RECTANGLE AND RHOMBUS COMBINED.
OTHER QUADRILATERALS
TRAPEZOIDS
• ONE PAIR OF PARALLEL SIDES
• BASES
• ISOSCELES HAVE CONGRUENT BASE ANGLES (2
PAIR) AND 2 CONGRUENT SIDES
KITES
• TWO PAIRS OF CONSECUTIVE SIDES CONGRUENT
• NO OPPOSITE SIDES CONGRUENT
• DIAGONALS ARE PERPENDICULAR
SIMILARITY
• CORRESPONDING ANGLES ARE CONGRUENT
• CORRESPONDING SIDES ARE PROPORTIONAL
• SCALE FACTOR = THE RATIO OF SIMILAR FIGURES
• SIMILARITY STATEMENT SHOWS CONGRUENT ANGLES, AND PROPORTIONAL SIDES (EXTENDED RATIO)
• SIMILAR SYMBOL IS ~
RIGHT TRIANGLES
PYTHAGOREAN THEOREM
• 𝐴2 + 𝐡2
= 𝐢2
• “A” IS A LEG
• “B” IS A LEG
• “C” IS THEY HYPOTENUSE (LONGEST SIDE)
PYTHAGOREAN TRIPLES
• WHOLE NUMBERS THAT SATISFY THE
PYTHAGOREAN THEOREM
• EXAMPLES INCLUDE: 3,4,5 AND 6,8,10.
• NO DECIMALS
• NO FRACTIONS
SPECIAL RIGHT TRIANGLES
30-60-90
45-45-90
• β„Žπ‘¦π‘π‘œπ‘‘π‘’π‘›π‘’π‘ π‘’ = 𝑙𝑒𝑔 2
• BOTH LEGS ARE THE SAME EXACT MEASURE.
• IF GIVEN A HYPOTENUSE, USE THE FOLLOWING
EQUATION TO SOLVE FOR THE LEG: 𝑙𝑒𝑔 =
β„Žπ‘¦π‘ 2
2
• β„Žπ‘¦π‘π‘œπ‘‘π‘’π‘›π‘’π‘ π‘’ = π‘ β„Žπ‘œπ‘Ÿπ‘‘ βˆ™ 2
• π‘™π‘œπ‘›π‘” 𝑙𝑒𝑔 = π‘ β„Žπ‘œπ‘Ÿπ‘‘ 3
• THERE ARE TWO WAYS TO FIND THE SHORT LEG IF
IT IS MISSING:
• π‘ β„Žπ‘œπ‘Ÿπ‘‘ = β„Žπ‘¦π‘π‘œπ‘‘π‘’π‘›π‘’π‘ π‘’
2
• π‘ β„Žπ‘œπ‘Ÿπ‘‘ =
• REMEMBER TO REDUCE ALL FRACTIONS.
π‘™π‘œπ‘›π‘” 3
3
TRIGONOMETRIC RATIOS
SINE
COSINE
TANGENT
π‘œπ‘π‘π‘œπ‘ π‘–π‘‘π‘’
β„Žπ‘¦π‘π‘œπ‘‘π‘’π‘›π‘’π‘ π‘’
π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘
β„Žπ‘¦π‘π‘œπ‘‘π‘’π‘›π‘’π‘ π‘’
π‘œπ‘π‘π‘œπ‘ π‘–π‘‘π‘’
π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘
AREA
• PARALLELOGRAMS = π‘π‘Žπ‘ π‘’ 𝑋 β„Žπ‘’π‘–π‘”β„Žπ‘‘
• SQUARES = 𝑠𝑖𝑑𝑒 2
• RECTANGLES = π‘™π‘’π‘›π‘”π‘‘β„Ž 𝑋 π‘€π‘–π‘‘π‘‘β„Ž
• TRIANGLES = ½ 𝑋 π‘π‘Žπ‘ π‘’ 𝑋 β„Žπ‘’π‘–π‘”β„Žπ‘‘
• CIRCLES = πœ‹π‘Ÿ2 R STANDS FOR RADIUS, AND A RADIUS ½ THE DIAMETER.
PERIMETER/CIRCUMFERENCE
• OF POLYGONS: SIMPLY ADD ALL THE SIDES!
• OF CIRCLES: 2πœ‹π‘Ÿ π‘œπ‘Ÿ πœ‹π‘‘ RADIUS IS THE “R”, DIAMETER IS THE “D”
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