4-4, 4-5 Congruent Triangles.ppt
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The document discusses different methods for proving that two triangles are congruent: - SSS (three pairs of corresponding sides are congruent) - SAS (two pairs of corresponding sides and the included angle are congruent) - ASA, AAS (two pairs of corresponding angles and one included or non-included side are congruent) - HL (hypotenuse-leg theorem for right triangles, showing hypotenuse and one leg are congruent) The document provides examples of using these methods like SAS, AAS, and ASA to prove triangles congruent through labeled diagram proofs. It emphasizes setting up the proof by marking congruent parts given and concluding that the triangles are congruent.
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Congruent triangles
Es una presentación del Profesor Jim Smith sobre congruencia de triángulos muy didáctica. Viene en idioma inglés.
Triangles and Types of triangles&Congruent Triangles (Congruency Rule)
This document defines and describes different types of triangles:
- Equilateral triangles have three equal sides and three equal angles.
- Isosceles triangles have at least two equal sides.
- Scalene triangles have no equal sides.
- Right triangles have one 90 degree angle.
- Acute triangles have all angles less than 90 degrees.
- Obtuse triangles have one angle greater than 90 degrees.
It also describes three theorems used to prove triangle congruence: SSS (three equal sides), SAS (two equal sides and the included angle), and ASA (two equal angles and one included side).
3-MATH 8-Q3-WEEK 2-ILLUSTRATING TRIANGLE CONGRUENCE AND Illustrating SSS, SAS...
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This document provides instructions and examples for illustrating triangle congruence. It begins with an activity asking students to identify whether figure pairs are congruent or not. Next, it discusses how to pair corresponding vertices, sides, and angles of congruent triangles. Examples are given demonstrating this process. The document then discusses different postulates for triangle congruence including SSS, SAS, and ASA. It provides additional examples and activities applying these postulates. It also discusses right triangle congruence and the corresponding theorems. In all, the document aims to teach students how to determine if two triangles are congruent and explain why using appropriate triangle congruence rules and terminology.4.3-5 Triangle Congruence
The document discusses different methods for proving triangles are congruent: SSS, SAS, ASA, AAS, and HL. It states that if the specified corresponding parts of two triangles are congruent using one of these methods, then the triangles are congruent. The document provides examples of congruence proofs using these methods and their reasoning.
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This document provides an introduction to congruent triangles and the different methods to prove triangles are congruent: SSS, SAS, and ASA. It includes examples of using side and angle correspondences to show triangles are congruent according to the three congruence rules. Students are asked to complete a KWL chart on triangles as an exit ticket.
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This document provides information about congruent triangles. It defines congruent triangles as two triangles that have the same shape and size, with corresponding sides and angles being equal. It describes several triangle congruence theorems including SSS, SAS, ASA, AAS, and RHS, which establish that triangles are congruent if certain combinations of sides and/or angles are equal. It also discusses isosceles triangles, angle bisectors, and provides examples applying the congruence theorems to prove triangles are congruent or not.
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This document discusses triangles and similarity. It provides examples of using the AAA, SAS, and SSS similarity criteria to determine if two triangles are similar. It also contains exercises involving applying similarity rules to pairs of triangles and finding missing angle measures. One exercise involves showing two triangles are similar using the fact that corresponding angles are equal for diagonals intersecting in a trapezium where the bases are parallel. The summary is provided in 3 sentences or less as requested.
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This document discusses triangles and similarity. It provides examples of using the AAA, SAS, and SSS similarity criteria to determine if two triangles are similar. It also contains exercises involving applying similarity rules to pairs of triangles and finding missing angle measures. One exercise involves showing two triangles are similar using the fact that corresponding angles are equal for diagonals intersecting in a trapezium where the bases are parallel. The summary is provided in 3 sentences or less as requested.
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Es una presentación del Profesor Jim Smith sobre congruencia de triángulos muy didáctica. Viene en idioma inglés.
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This document defines and describes different types of triangles:
- Equilateral triangles have three equal sides and three equal angles.
- Isosceles triangles have at least two equal sides.
- Scalene triangles have no equal sides.
- Right triangles have one 90 degree angle.
- Acute triangles have all angles less than 90 degrees.
- Obtuse triangles have one angle greater than 90 degrees.
It also describes three theorems used to prove triangle congruence: SSS (three equal sides), SAS (two equal sides and the included angle), and ASA (two equal angles and one included side).
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This document provides instructions and examples for illustrating triangle congruence. It begins with an activity asking students to identify whether figure pairs are congruent or not. Next, it discusses how to pair corresponding vertices, sides, and angles of congruent triangles. Examples are given demonstrating this process. The document then discusses different postulates for triangle congruence including SSS, SAS, and ASA. It provides additional examples and activities applying these postulates. It also discusses right triangle congruence and the corresponding theorems. In all, the document aims to teach students how to determine if two triangles are congruent and explain why using appropriate triangle congruence rules and terminology.4.3-5 Triangle Congruence
The document discusses different methods for proving triangles are congruent: SSS, SAS, ASA, AAS, and HL. It states that if the specified corresponding parts of two triangles are congruent using one of these methods, then the triangles are congruent. The document provides examples of congruence proofs using these methods and their reasoning.
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This document provides an introduction to congruent triangles and the different methods to prove triangles are congruent: SSS, SAS, and ASA. It includes examples of using side and angle correspondences to show triangles are congruent according to the three congruence rules. Students are asked to complete a KWL chart on triangles as an exit ticket.
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1. SSS (side-side-side)
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The document discusses different methods for proving triangles congruent:
1. SSS (side-side-side)
2. SAS (side-angle-side)
3. ASA (angle-side-angle)
4. AAS (angle-angle-side)
5. HL (hypotenuse-leg) for right triangles only.
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The document discusses different methods for proving triangles congruent:
1. SSS (side-side-side)
2. SAS (side-angle-side)
3. ASA (angle-side-angle)
4. AAS (angle-angle-side)
5. HL (hypotenuse-leg) for right triangles only.
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2. If two shapes are similar, one is an enlargement of the other. The scale factor represents the ratio of corresponding sides.
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This document provides information and formulas for calculating the perimeter and area of various polygons. It includes formulas for finding the perimeter and area of rectangles, parallelograms, triangles, trapezoids, rhombi, kites, and regular polygons. The key points are that perimeter is found by adding all the sides of any polygon, and area formulas vary depending on the specific shape but generally involve multiplying base and height or base and apothem.
10-8 Equations of circles.ppt
The document discusses finding the equation of a circle given its center and radius, or finding the center and radius given the equation. It provides examples of writing the standard equation of a circle (x - h)2 + (y - k)2 = r2 given the center (h, k) and radius r. It also works through examples of finding the center and radius when given the equation of a circle. Finally, it explains how to find the equation of a circle if given its center and that it passes through a specific point.
10-7 Circles - Segments.ppt
This document contains instructions and examples for measuring circles, segments, chords, and tangents. It includes the following examples:
1) Finding the length of a chord using the formula that the product of the parts of one chord equals the product of the parts of the other.
2) Solving a pair of equations involving a segment, radius, and variable part of a chord.
3) Explaining that the diameter of a circle bisects any chord perpendicular to it.
The document provides step-by-step workings for each example and problem.
10-4, 10-6 Circles and Angles.ppt
This document discusses properties of angles related to circles. It defines central angles, inscribed angles, and their relationships to intercepted arcs. The measure of a central angle is equal to the measure of its intercepted arc. The measure of an inscribed angle is equal to half the measure of its intercepted arc. Examples are given of finding the measures of unknown angles using these properties.
10-3_Arcs_and_Chords.ppt
This document discusses properties of chords and arcs in circles. It states that if two chords are congruent, their corresponding arcs are also congruent. It also notes that equal chords are equidistant from the center, and chords get longer as they get closer to the center, with diameters being the longest chords that pass through the center. The document provides an example of how to find the center of a circle by drawing two chords and finding the intersection of the perpendicular bisectors.
10-2_Angles_and_Arcs.ppt
This document discusses angles and arcs in circles. An angle formed by two radii is called a central angle. The measure of an arc is equal to the measure of the central angle. There are three types of arcs: minor arcs which are named by their endpoints, major arcs which are named by endpoints and go through another point, and semicircles which are arcs that are equal to 180 degrees. The document also provides an example of using percentages to find the measures of central angles for different sports in a pie chart.
10-1 Circles.ppt
The document defines and describes the key parts of a circle, including the center, radius, diameter, chord, and circumference. It provides formulas for calculating the radius, diameter, and circumference based on these parts. Examples are given for using the formulas to calculate lengths and circumferences of various circles.
8-7 Law of Cosines.ppt
The document discusses the Law of Cosines, which can be used to find the angles of a triangle when only the three sides are known. It provides the Law of Cosine formulas for finding the angles opposite sides a, b, and c. Examples are given for finding a missing side with two sides and the included angle (SAS) and for finding a missing angle with all three sides known (SSS). A word problem example finds the distance a golf ball is from the cup using the Law of Cosines.
8-6 Law of Sines.ppt
The document discusses the Law of Sines and how to use it to solve for missing sides and angles in non-right triangles. It provides examples of using the Law of Sines to find a missing side length given two angles and one side, and to find a missing angle given two sides and the angle between them. Finally, it applies the Law of Sines to solve a real-world problem of finding the height of a telephone pole given the angle of elevation to its top, its shadow length, and the angle the pole was knocked from perpendicular.
8-5 Trig Angles of Elev and Depress.ppt
This document defines and provides examples of angles of elevation and depression. It discusses how these angles are formed when viewing an object from a higher or lower position and must have one horizontal side. Several word problems are worked out using trigonometric functions like sine, tangent, and inverse tangent to calculate unknown distances and heights based on a given angle of elevation or depression and one known side length.
8-4 Sine, Cosine.ppt
The document defines the trigonometric functions of sine, cosine, and tangent. It provides examples of calculating unknown side lengths and angles using trigonometric ratios and the definitions of sine, cosine, and tangent. Sample calculations include finding an unknown side length given two other sides and an angle measure, and finding an unknown angle measure given two side lengths.
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- 1. Jim Smith JCHS Sections 4-3, 4-5 3108.4.38 Use the principle that corresponding parts of congruent triangles are congruent to solve problems
- 2. When we talk about congruent triangles, we mean everything about them is congruent. All 3 pairs of corresponding angles are equal…. And all 3 pairs of corresponding sides are equal
- 3. For us to prove that 2 people are identical twins, we don’t need to show that all “2000” body parts are equal. We can take a short cut and show 3 or 4 things are equal such as their face, age and height. If these are the same I think we can agree they are twins. The same is true for triangles. We don’t need to prove all 6 corresponding parts are congruent. We have 5 short cuts or methods.
- 4. SSS If we can show all 3 pairs of corr. sides are congruent, the triangles have to be congruent.
- 5. SAS Show 2 pairs of sides and the included angles are congruent and the triangles have to be congruent. Included angle Non-included angles
- 6. This is called a common side. It is a side for both triangles. We’ll use the reflexive property.
- 7. Which method can be used to prove the triangles are congruent
- 8. Common side SSS Parallel lines alt int angles Common side Vertical angles SAS SAS
- 10. ASA, AAS and HL ASA – 2 angles and the included side A S A AAS – 2 angles and The non-included side A A S
- 11. HL ( hypotenuse leg ) is used only with right triangles, BUT, not all right triangles. HL ASA
- 12. When Starting A Proof, Make The Marks On The Diagram Indicating The Congruent Parts. Use The Given Info, Properties, Definitions, Etc. We’ll Call Any Given Info That Does Not Specifically State Congruency Or Equality A PREREQUISITE PROOFS
- 13. SOME REASONS WE’LL BE USING • DEF OF MIDPOINT • DEF OF A BISECTOR • VERT ANGLES ARE CONGRUENT • DEF OF PERPENDICULAR BISECTOR • REFLEXIVE PROPERTY (COMMON SIDE) • PARALLEL LINES ….. ALT INT ANGLES
- 14. A B C D E 1 2 Given: AB = BD EB = BC Prove: ∆ABE ˜ ∆DBC = SAS Our Outline P rerequisites S ides A ngles S ides Triangles ˜ =
- 15. A C D Given: AB = BD EB = BC Prove: ∆ABE ˜ ∆DBC = B E 1 2 SAS none AB = BD Given 1 = 2 Vertical angles EB = BC Given ∆ABE ˜ ∆DBC SAS = STATEMENTS REASONS P S A S ∆’s
- 16. A B C 1 2 Given: CX bisects ACB A ˜ B Prove: ∆ACX ˜ ∆BCX X = = AAS P A A S ∆’s CX bisects ACB Given 1 = 2 Def of angle bisc A = B Given CX = CX Reflexive Prop ∆ACX ˜ ∆BCX AAS =
- 17. Can you prove these triangles are congruent? A B D C X Given: AB ll DC X is the midpoint of AC Prove: AXB ˜ CXD =
- 18. A B D C X Given: AB ll DC X is the midpoint of AC Prove: AXB ˜ CXD = AAS P A A S ∆’s AB ll DC , X is midpt of AC GIVEN AX = XC AAS
- 19. A B D C X Given: AB ll DC X is the midpoint of AC Prove: AXB ˜ CXD = ASA A B D C X AAS