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Terms in this set (25)
Four Properties of Proportions
1 ▶︎▶︎▶︎ MEANS-EXTREMES PROPERTY, or CROSS-PRODUCTS PROPERTY (also known as Property 1):
➜ If a/b = c/d, then a • d = b • c.
➜ Conversely, if a • d = b • c, and neither a • d nor b • c equals zero, then a/b = c/d and b/a = d/c
For example, for the proportion 8/10 = 4/5,
the Means-Extremes Property (Property 1) specifies 8 • 5 = 10 • 4, or 40 = 40.
2 ▶︎▶︎▶︎ MEANS OR EXTREMES SWITCHING PROPERTY (also known as Property 2):
➜ If a/b = c/d and is a proportion, then both d/b = c/a and a/c = b/d are proportions.
For example, for the proportion 8/10 = 4/5,
the Means or Extremes Switching Property (Property 2) specifies that if you were to switch the 8 and 5 or switch the 4 and 10, then the new statement is still an accurate proportion.
If 8/10 = 4/5, then 5/10 = 4/8, OR if 8/10 = 4/5, then 8/4 = 10/5.
3 ▶︎▶︎▶︎ UPSIDE-DOWN PROPERTY (also known as Property 3):
➜ If a/b = c/d, then b/a = d/c.
For example, if 9 • a = 5 • b, and the product ≠ 0, then find the ratio for a/b.
First, apply the converse of the Cross Products Property and obtain 9/5 = b/a.
Next, proceed in one of the following two ways:
☛ Apply Property 3 to 9/5 = b/a:
Turn each side upside-down.
5/9 = a/b, or a/b = 5/9
☛ Apply Property 2 to 9/b = 5/a:
Switch the 9 and the a, so that a/b = 5/9
4 ▶︎▶︎▶︎ DENOMINATOR ADDITION/SUBTRACTION PROPERTY (also known as Property 4):
➜ If a/b = c/d, then (a + b)/ b = ( c + d)/ d or (a − b)/ b = (c − d)/ d.
Copy and paste the following link into your browser to learn more about using the four properties of proportion in geometry:
https://youtu.be/fvtlFmRA5sY
Similar Polygons
Two polygons with the same shape are called SIMILAR POLYGONS. The symbol for "is similar to" is ∼. Notice that it is a portion of the "is congruent to" symbol, ≅. When two polygons are similar, these two facts BOTH must be true:
➜ Corresponding angles are equal.
➜ The ratios of pairs of corresponding sides must all be equal.
Two geometrical objects are called SIMILAR if they both have the same shape, or one has the same shape as the mirror image of the other.
More precisely, one can be obtained from the other by uniformly scaling (enlarging or reducing), possibly with additional translation, rotation and reflection.
This means that either object can be rescaled, repositioned, and reflected, so as to coincide precisely with the other object.
If two objects are similar, each is CONGRUENT to the result of a particular uniform scaling of the other. A modern and novel perspective of similarity is to consider geometrical objects similar if one appears congruent to the other when zoomed in or out at some level.
For example, all circles are similar to each other, all squares are similar to each other, and all equilateral triangles are similar to each other.
On the other hand, ellipses are NOT all similar to each other, rectangles are NOT all similar to each other, and isosceles triangles are NOT all similar to each other.
Copy and paste the following link into your browser to learn more about using similar polygons in geometry:
https://youtu.be/AFEDUm4bPHk
Similar Triangles and AA (Angle-to-Angle) Similarity Postulate
In general, to prove that two polygons are similar, you must show that all pairs of corresponding angles are equal and that all ratios of pairs of corresponding sides are equal. In triangles, though, this is NOT necessary.
Postulate 17 (AA Similarity Postulate) specifies, if two angles of one triangle are equal to two angles of another triangle, then the triangles are similar.
Copy and paste the following link into your browser to learn more about working with similar triangles and the AA similarity postulate in geometry:
https://youtu.be/k_iAz3_lE_w
Proportional Parts of Triangles, Segment Addition Postulate and Side‐Splitter Theorem
▶︎▶︎▶︎ PROPORTIONAL PARTS IN TRIANGLES:
If a line is drawn in a triangle so that it is parallel to one of the sides and it intersects the other two sides then the segments are of proportional lengths:
Parts of two triangles can be proportional; if two triangles are known to be similar then the perimeters are proportional to the measures of corresponding sides.
▶︎▶︎▶︎ SEGMENT ADDITION POSTULATE:
In geometry, the segment addition postulate states that given 2 points A and C, a third point B lies on the line segment AC if and only if the distances between the points satisfy the equation AB + BC = AC.
▶︎▶︎▶︎ The SIDE SPLITTER THEOREM states that if a line is parallel to a side of a triangle and intersect the other two sides, then this line divides those two sides proportionally.
The side splitter theorem is a natural extension of similarity ratio, and it happens any time that a pair of parallel lines intersect a triangle.
Copy and paste the following link into your browser to learn more about using proportional parts of triangles in geometry:
https://youtu.be/Z8X-bdyumZc
Proportional Parts of Similar Triangles
According to the following theorem for similar triangles:
➜ THEOREM 59:
If two triangles are similar, then the ratio of any two corresponding segments (such as altitudes, medians, or angle bisectors) equals the ratio of any two corresponding sides
When we say we have two triangles which are similar, it means that the CORRESPONDING SIDES of the two triangles are in PROPORTION.
The ratio of corresponding parts of the similar triangles always gives the same value for all the three sides.
When we talk about the two similar Δ's it means that the two triangles looks same but the measure of the line segment of the two triangles is not same.
Copy and paste the following link into your browser to learn more about using proportional parts of similar triangles in geometry:
https://www.bing.com/videos/search?q=using+proportional+parts+of+similar+triangles+in+geometry&&view=detail&mid=D7EA473423AC16AEFA86D7EA473423AC16AEFA86&FORM=VRDGAR
Ratios and Proportions; Means and Extremes
▶︎▶︎▶︎ RATIO
The RATIO of two numbers 'a' and 'b' is the fraction a/b , usually expressed in reduced form. An alternative form involves a colon. [a:b]. The colon form is most frequently used when comparing three or more numbers to each other.
▶︎▶︎▶︎ PROPORTION
A PROPORTION is an equation stating that two ratios are equal. For example, ½ = 3/6
▶︎▶︎▶︎ MEANS AND EXTREMES
The EXTREMES are the terms in a proportion that are the farthest apart when the proportion is written in colon form ( a:b = c:d). In the foregoing, a and d are extremes. The MEANS are the two terms closest to each other. For example,
a/b = c/d, or a : b = c : d, where 'a' and 'd' are the EXTREMES, while 'b' and 'c' are the MEANS
In the preceding proportion, the values a and d are called extremes of the proportion; the values b and c are called the means of the proportion
Similar Triangles: Perimeters and Areas and Scale Factor
When two triangles are similar, the reduced ratio of any two corresponding sides is called the SCALE FACTOR OF THE SIMILAR TRIANGLES.
Multiplying the smaller triangle's side lengths by the scale factor will give you the side lengths of the larger triangle. Similarly, dividing the larger triangle's side lengths by the scale factor will give you the side lengths of the smaller triangle.
These theorems are related to the scale factor of similar triangles:
➜ THEOREM 60:
If two similar triangles have a scale factor of a : b, then the ratio of their perimeters is a : b.
➜ THEOREM 61:
If two similar triangles have a scale factor of a : b, then the ratio of their areas is a² : b².
Copy and paste the following link into your browser to learn more about perimeters, areas and scale factor of similar triangles:
https://www.bing.com/videos/search?q=perimeters%2c+areas+and+scale+factor+of+similar+triangles&&view=detail&mid=DEA870FB5BA3F79668BEDEA870FB5BA3F79668BE&FORM=VRDGAR
Segment Addition Postulate
According to the SEGMENT ADDITION POSTULATE, point B is a point on segment AC, that is, B is between A and C, if and only if AB + BC
In other words, if three points A, B, and C are collinear and B is between A and C, then AB + BC = AC
Similarity: Side-Side-Side (SSS) Congruence
SIDE-SIDE-SIDE (SSS) CONGRUENCE specifies, if three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.
Similarity: Side-Angle-Side (SAS) Congruence
SIDE-ANGLE-SIDE (SAS) CONGRUENCE
specifies, if two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.
Similarity: Angle-Side-Angle (ASA) Congruence
ANGLE-SIDE-ANGLE (ASA) CONGRUENCE specifies if two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.
Similarity: Angle-Angle-Side (AAS) Congruence
ANGLE-ANGLE-SIDE (AAS) CONGRUENCE specifies if two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.
Similarity: Hypotenuse-Leg (HL) Congruence
HYPOTENUSE-LEG (HL) CONGRUENCE (right triangle) specifies if the hypotenuse and leg of one right triangle are congruent to the corresponding parts of another right triangle, the two right triangles are congruent.
Similarity: CPCTC
CPCTC represents: Corresponding parts of congruent triangles are congruent.
Similarity: Angle-Angle (AA) Similarity
ANGLE-ANGLE (AA) SIMILARITY specifies if two angles of one triangle are congruent to two angles of another triangle, the triangles are similar.
Similarity: SSS for Similarity
SSS FOR SIMILARITY specifies if the three sets of corresponding sides of two triangles are in proportion, the triangles are similar.
Similarity: Side Proportionality
SAS FOR SIMILARITY specifies if an angle of one triangle is congruent to the corresponding angle of another triangle and the lengths of the sides including these angles are in proportion, the triangles are similar.
Similarity: Side Proportionality
SIDE PROPORTIONALITY specifies if two triangles are similar, the corresponding sides are in proportion.
Similarity: Mid-Line Theorem
MID-SEGMENT THEOREM (also called mid-line) specifies that the segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long.
Similarity: Sum of Two Sides
Sum of Two Sides
The sum of the lengths of any two sides of a triangle must be greater than the third side
Similarity: Longest Side
LONGEST SIDE specifies that in a triangle, the longest side is across from the largest angle. In a triangle, the largest angle is across from the longest side.
Similarity: Altitude Rule
ALTITUDE RULE specifies that the altitude to the hypotenuse of a right triangle is the mean proportional between the segments into which it divides the hypotenuse..
Similarity: Leg Rule
LEG RULE specifies that each leg of a right triangle is the mean proportional between the hypotenuse and the projection of the leg on the hypotenuse.
Similarity: Base Angle Theorem (Isosceles Triangle)
BASE ANGLE THEOREM (ISOSCELES TRIANGLE) specifies if two sides of a triangle are congruent, the angles opposite these sides are congruent.
Similarity: Base Angle Converse (Isosceles Triangle)
BASE ANGLE CONVERSE (ISOSCELES TRIANGLE) specifies if two angles of a triangle are congruent, the sides opposite these angles are congruent.
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