Chapter 2 Similarity and Congruence
Definitions Definition AB = CD if and only if AB = CD Remember, mab = AB.
Definitions Definition AB = CD if and only if AB = CD Remember, mab = AB. Definition ABC = DEF if and only if m ABC = m DEF
Congruence Postulates Postulate Side-Side-Side If three sides of a triangle are congruent to the corresponding sides in another triangle, then the triangles are congruent.
Congruence Postulates Postulate Side-Angle-Side If two sides and the included angle of a triangle are congruent to the corresponding sides and angle in another triangle, then the triangles are congruent.
Congruence Postulates Postulate Hypotenuse-Leg If the hypotenuse and leg of one right triangle are congruent to the hypotenuse and corresponding leg of another triangle, then the two triangles are congruent.
Euclidean Tools A compass looks like our compass, but it has no markings on it. So we can t set it to draw circles with predetermined radii. Also, when we pick up the compass, it collapses, so we cannot copy a circle by picking up the compass and drawing another. We can only draw circles when given a center and a point on the circumference.
Euclidean Tools A compass looks like our compass, but it has no markings on it. So we can t set it to draw circles with predetermined radii. Also, when we pick up the compass, it collapses, so we cannot copy a circle by picking up the compass and drawing another. We can only draw circles when given a center and a point on the circumference. The straightedge is like a ruler with no markings. We can make straight lines as long as we choose using any two points, or we can extend an existing line segment as long as we want.
What We Can Construct Create a line through two points
What We Can Construct Create a line through two points Create a point at the intersection of lines
What We Can Construct Create a line through two points Create a point at the intersection of lines Question What is the difference between intersecting and concurrent lines?
What We Can Construct Question Create a line through two points Create a point at the intersection of lines What is the difference between intersecting and concurrent lines? Create a circle with two points where one is the center and the other is any point on the circumference
What We Can Construct Question Create a line through two points Create a point at the intersection of lines What is the difference between intersecting and concurrent lines? Create a circle with two points where one is the center and the other is any point on the circumference Create point(s) of intersection of lines and circles Create point(s) of intersection of two circles
What We Can Construct Question Create a line through two points Create a point at the intersection of lines What is the difference between intersecting and concurrent lines? Question Create a circle with two points where one is the center and the other is any point on the circumference Create point(s) of intersection of lines and circles Create point(s) of intersection of two circles For these two, how many points could there be? For the intersection of lines, circles or one of each, how many points of intersection could there be?
What We Can Construct Bisect an angle
What We Can Construct Bisect an angle Find the perpendicular bisector of a segment
What We Can Construct Bisect an angle Find the perpendicular bisector of a segment Construct regular polygons
What We Can Construct Bisect an angle Find the perpendicular bisector of a segment Construct regular polygons Circumscribe regular polygons
What We Can Construct Bisect an angle Find the perpendicular bisector of a segment Construct regular polygons Circumscribe regular polygons Circumscribe some other polygons
Terms We ll Need Definition A perpendicular bisector is a line that passes through the midpoint of another line segment and the intersection forms four right angles.
Terms We ll Need Definition A perpendicular bisector is a line that passes through the midpoint of another line segment and the intersection forms four right angles. Definition The altitude of a triangle is a line segment that begins at a vertex of a triangle and is perpendicular to the opposite side.
Terms We ll Need Definition A perpendicular bisector is a line that passes through the midpoint of another line segment and the intersection forms four right angles. Definition The altitude of a triangle is a line segment that begins at a vertex of a triangle and is perpendicular to the opposite side. Definition An angle bisector is a line that passes through the vertex of an angle and divides the angle into two equal angles.
Circumscribing Polygons Definition Circumscribing a polygon means we draw a circle that passes through all of the vertices of the polygon.
Circumscribing Polygons Definition Circumscribing a polygon means we draw a circle that passes through all of the vertices of the polygon. Definition The point at which the perpendicular bisectors of the sides of a triangle meet is the circumcenter. The circle we draw that passes through each vertex is called the circumcircle.
When Can We Circumscribe a Quadrilateral? Theorem a. If a circle can be circumscribed about a convex quadrilateral, then the opposite angles are supplementary. b. If the opposite angles of a quadrilateral are supplementary, then a circle can be circumscribed about the quadrilateral.
ASA Postulate Angle-Side-Angle If two angles and the included side of one triangle are congruent to two two angles and the included side in another triangle, respectively, then the triangles are congruent.
AAS Postulate Angle-Angle-Side If two angles and a side opposite one of these angles of a triangle are congruent to two two angles and the corresponding side in another triangle, then the triangles are congruent.
SSA Postulate Side-Side-Angle Postulate This postulate doesn t exist The question is, why?
The Postulates at Work Example Given that PQ RS and PRQ = SQR, prove that PQR = SRQ. But first, why did I have to give you that PRQ = SQR instead of just telling you PQ RS?
The Postulates at Work Example Given that RN bisects ERV and NER = NVR, prove that ENR = VNR.
The Postulates at Work Example If AM and BN bisect the base angles of the given isosceles triangle, prove AM = BN. C N M A B
What We Want To Construct 1 Parallel lines 2 Parallelograms 3 Perpendicular line to a given point 4 Angles other than 90 5 Incenter
Definition Definition The altitude of a triangle is the perpendicular from the base to the opposite vertex.
Angle Bisectors Theorem a. Any point P on an angle bisector is equidistant from the sides of the angle. b. Any point in the interior of an angle that is equidistant from the sides of the angle is on the angle bisector of the angle.
Incenter Definition The incenter of a triangle is the point of concurrency for the angle bisectors of a triangle.
Incenter Definition The incenter of a triangle is the point of concurrency for the angle bisectors of a triangle. Theorem The incenter of a triangle is equidistant from the three sides of the triangle.
Definition of Similarity Definition ABC is similar to DEF, denoted as ABC DEF, if and only if the corresponding angles are congruent and the corresponding sides are proportional. A D 4 6 2 E 2 3 F B 4 C AB DE = BC EF = AC DF
Question Explain the following: Are all isosceles triangles similar?
Ways to Prove Similarity of Triangles Theorem SSS Similarity for Triangles If the lengths of corresponding sides of two triangles are proportional, then the triangles are similar.
Ways to Prove Similarity of Triangles Theorem SAS Similarity for Triangles If two sides are proportional to the corresponding sides and the included angles are congruent, then the triangles are similar.
Ways to Prove Similarity of Triangles Theorem AA Triangle Similarity If two angles in one triangle are congruent to the corresponding angles in another triangle, then the triangles are similar.
Example Triangle Similarity Explain why DBE ABC. What is the length of BE?
Example Triangle Similarity Explain why DBE ABC. What is the length of BE? 4 12 = x x + 9 8x = 36 x = 9 2
Example Triangle Similarity Explain why ABC ADB. Find the value of x.
Solution x 3 = x + 4 6 6x = 3(x + 4) 6x = 3x + 12 3x = 12 x = 4
Solution x 3 = x + 4 6 6x = 3(x + 4) 6x = 3x + 12 3x = 12 x = 4 Notice now that the length of the side AD is twice the length of AB, giving us a ratio of 1 2 for the measures of the sides in ABC compared to the corresponding sides of ADE.
Theorem Theorem If a line parallel to one side of a triangle intersects the other sides then it divides those sides into proportional segments.
Theorem Theorem If a line parallel to one side of a triangle intersects the other sides then it divides those sides into proportional segments. Theorem If a line divides two sides of a triangle into proportional segments, then the line is parallel to the third side.
Theorem Theorem If a line parallel to one side of a triangle intersects the other sides then it divides those sides into proportional segments. Theorem If a line divides two sides of a triangle into proportional segments, then the line is parallel to the third side. Theorem If a parallel line cuts off congruent segments on one transversal, then they cut off congruent segments on any transversal.
Midpoints Definition The midsegment is the segment connecting the midpoint of adjacent sides of a triangle or quadrilateral.
Midpoints Definition The midsegment is the segment connecting the midpoint of adjacent sides of a triangle or quadrilateral. Theorem The Midpoint Theorem The midsegment joining the midpoint of two sides of a triangle is parallel to and is half as long as the third side.
Midpoints Definition The midsegment is the segment connecting the midpoint of adjacent sides of a triangle or quadrilateral. Theorem The Midpoint Theorem The midsegment joining the midpoint of two sides of a triangle is parallel to and is half as long as the third side. Theorem If a line bisects one side of a triangle and is parallel to a second side then it bisects the third side and therefore is a midsegment.
Centroid Definition The median of a triangle is the segment joining a vertex and the midpoint of the opposite side.
Centroid Definition The median of a triangle is the segment joining a vertex and the midpoint of the opposite side. Definition The centroid is the point of concurrency of the three medians of a triangle.
Example Triangle Similarity Explain why abc fde
Example Similarity Find the value of z.
Example Similarity Find the value of z. 6 10 = 4 z z = 20 3
Example More Similarity Justify why these triangles are similar and then find the value of x and y.
Example More Similarity Justify why these triangles are similar and then find the value of x and y. 12 x = 16 20 = 20 y
Example More Similarity Justify why these triangles are similar and then find the value of x and y. So, x = 15 and y = 25. 12 x = 16 20 = 20 y
Similarity and Other Polygons Definition Any two polygons with the same number of sides are similar if and only if the corresponding angles are congruent and the corresponding sides are proportional.
Similarity and Other Polygons Definition Any two polygons with the same number of sides are similar if and only if the corresponding angles are congruent and the corresponding sides are proportional. Same idea without the named theorems and postulates.
Example Similarity Suppose you wanted to make a copy of a document at 1 8 of the original size, but you made a mistake and made a copy of the original at 2 5 of the original size. You are stubborn, so instead of starting at over, you want to use the copy you made and reduce it to make the final product be 1 8 of the original size. What ratio should you use to do this?
Example Similarity Suppose you wanted to make a copy of a document at 1 8 of the original size, but you made a mistake and made a copy of the original at 2 5 of the original size. You are stubborn, so instead of starting at over, you want to use the copy you made and reduce it to make the final product be 1 8 of the original size. What ratio should you use to do this? We think of this as 1 8 is the part we want and 2 5 is the whole, since that is what we are working with now. But we want to know what part of the original whole this corresponds to. This gives 1 8 2 5 = x 100
Example Similarity Suppose you wanted to make a copy of a document at 1 8 of the original size, but you made a mistake and made a copy of the original at 2 5 of the original size. You are stubborn, so instead of starting at over, you want to use the copy you made and reduce it to make the final product be 1 8 of the original size. What ratio should you use to do this? We think of this as 1 8 is the part we want and 2 5 is the whole, since that is what we are working with now. But we want to know what part of the original whole this corresponds to. This gives 1 8 2 5 = x 100 5 16 = x 100 16x = 500 x = 31.25
Similarity Example Suppose we have ABC, DEF, and GHI such that ABC is 70% of DEF and GHI is 30% of DEF. What is the ratio between ABC and GHI?
Similarity Example Suppose we have ABC, DEF, and GHI such that ABC is 70% of DEF and GHI is 30% of DEF. What is the ratio between ABC and GHI? We have AB DE = 70 100, GH DE = 30 100
Similarity Example Suppose we have ABC, DEF, and GHI such that ABC is 70% of DEF and GHI is 30% of DEF. What is the ratio between ABC and GHI? We have AB DE = 70 100, GH DE = 30 100 AB DE GH DE = 70 100 30 100
Similarity Example Suppose we have ABC, DEF, and GHI such that ABC is 70% of DEF and GHI is 30% of DEF. What is the ratio between ABC and GHI? We have AB DE = 70 100, GH DE = 30 100 AB DE GH DE = 70 100 30 100 ab gh = 70 30 So, the ratio between the first and third triangles is 7 3.
Similarity and Slope Do you remember the formula for slope? How about the phrase we use when working with slope?
Similarity and Slope Do you remember the formula for slope? How about the phrase we use when working with slope? Slope m = rise run = y 2 y 1 x 2 x 1
Similarity and Slope Do you remember the formula for slope? How about the phrase we use when working with slope? Slope m = rise run = y 2 y 1 x 2 x 1 How does this relate to similar triangles?