right triangles and similarity common core geometry homework answers

Common Core Geometry Math (Worksheets, Homework, Lesson Plans)

Related Topics: Common Core Math Resources, Lesson Plans & Worksheets for all grades Common Core Math Video Lessons, Math Worksheets and Games for Geometry Common Core Math Video Lessons, Math Worksheets and Games for all grades

Looking for video lessons that will help you in your Common Core Geometry math classwork or homework? Looking for Common Core Math Worksheets and Lesson Plans that will help you prepare lessons for Geometry students?

The following lesson plans and worksheets are from the New York State Education Department Common Core-aligned educational resources. Eureka/EngageNY Math Geometry Worksheets.

These Lesson Plans and Worksheets are divided into five modules.

Geometry Homework, Lesson Plans and Worksheets

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• About the Standards
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Kindergarten

• Counting & Cardinality
• Operations & Algebraic Thinking
• Number & Operations in Base Ten
• Measurement & Data
• Number & Operations—Fractions¹
• Number & Operations in Base Ten¹
• Number & Operations—Fractions
• Ratios & Proportional Relationships
• The Number System
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• The Real Number System
• Quantities*
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High School: Algebra

• Seeing Structure in Expressions
• Arithmetic with Polynomials & Rational Expressions
• Creating Equations*
• Reasoning with Equations & Inequalities

High School: Functions

• Interpreting Functions
• Building Functions
• Linear, Quadratic, & Exponential Models*
• Trigonometric Functions

High School: Modeling

High school: geometry.

• Similarity, Right Triangles, & Trigonometry
• Expressing Geometric Properties with Equations
• Geometric Measurement & Dimension
• Modeling with Geometry

High School: Statistics & Probability

• Interpreting Categorical & Quantitative Data
• Making Inferences & Justifying Conclusions
• Conditional Probability & the Rules of Probability
• Using Probability to Make Decisions

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• Courses & Transitions

Mathematics Glossary

Mathematics » high school: geometry » similarity, right triangles, & trigonometry, standards in this domain:, understand similarity in terms of similarity transformations.

• A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
• The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
• G-SRT.2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
• G-SRT.3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

Prove theorems involving similarity

• G-SRT.4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
• G-SRT.5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Define trigonometric ratios and solve problems involving right triangles

• G-SRT.6. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
• G-SRT.7. Explain and use the relationship between the sine and cosine of complementary angles.
• G-SRT.8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. ★

Apply trigonometry to general triangles

• G-SRT.9. (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
• G-SRT.10. (+) Prove the Laws of Sines and Cosines and use them to solve problems.
• G-SRT.11. (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).
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Common Core: High School - Geometry : Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures

Study concepts, example questions & explanations for common core: high school - geometry, all common core: high school - geometry resources, example questions, example question #1 : use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Choose the answer that describes a congruent triangle

Triangles with corresponding sides of the same lengths but differing corresponding angles

Triangles with corresponding angles of the same measure but differing corresponding sides

Triangles with corresponding sides of the same lengths and corresponding angles of the same measures

Triangles with 2 of the 3 corresponding sides of the same lengths but differing corresponding angles

To be congruent, triangles must have all corresponding angles and sides be of the same measure.  Corresponding sides of triangles are sides of the same measure in the same positions on different triangles.  This is shown in red on the two triangles below.  Corresponding angles of triangles are angles of the same measure in the same position on different triangles.  This is shown in blue on the two triangles below.

Example Question #2 : Use Congruence And Similarity Criteria For Triangles To Solve Problems And To Prove Relationships In Geometric Figures

Choose the answer that describes similar triangles

Shapes, in general, are similar when they are the same shape but have different sizes.  If they were the exact same size, they would be congruent.  To have the same shape, even when the sizes differ, the shape needs to have the same angles.  Take the two triangles below for example, they have equal angles but are different sizes.  Therefore these triangles are similar.

True or False: When considering right triangles, if two right triangles have a congruent hypotenuse and a congruent leg then these triangles are congruent.

If two right triangles have a congruent leg and hypotenuse we can say they have two congruent sides.  Since both of these triangles are right triangles, we also know that they have a congruent angle (their 90-degree angle).  So these two triangles have two pairs of corresponding congruent sides and one pair of corresponding congruent angles.  By SAS Theorem, these right triangles are congruent.  When using this fact that two right triangles are congruent when they have a congruent hypotenuse and a congruent leg, this is called the HL Theorem.

Example Question #4 : Use Congruence And Similarity Criteria For Triangles To Solve Problems And To Prove Relationships In Geometric Figures

True or False: Triangles that have all three corresponding sides equal in length can still have differing corresponding angles, therefore triangles with all three corresponding sides equal in length are not guaranteed to be either similar or congruent.

The Side-Side-Side (SSS) Theorem states that two triangles are congruent if the three corresponding sides of each triangle are congruent.  If you think about it, in order for all three corresponding sides to be equal they must fit together at the same angles as well making these triangles congruent.  A detailed proof based on Philoh’s approach is below for a better explanation.  Note that there are stronger proofs than this for this theorem, but this proof is best visually for this level of understanding of the theorem.

(We want to show that if all three sides of two triangles are equal, then these two triangles are congruent)

Example Question #5 : Use Congruence And Similarity Criteria For Triangles To Solve Problems And To Prove Relationships In Geometric Figures

Explanation: For the explanation follow the detailed proof below:

Example Question #6 : Use Congruence And Similarity Criteria For Triangles To Solve Problems And To Prove Relationships In Geometric Figures

So we can set up the following ratios

Example Question #7 : Use Congruence And Similarity Criteria For Triangles To Solve Problems And To Prove Relationships In Geometric Figures

No relationship

Example Question #8 : Use Congruence And Similarity Criteria For Triangles To Solve Problems And To Prove Relationships In Geometric Figures

There is not enough information to determine if these triangles are similar

No, these triangles are not similar

We are able to set up the following ratios:

Example Question #10 : Use Congruence And Similarity Criteria For Triangles To Solve Problems And To Prove Relationships In Geometric Figures

True or False: The triangles below are similar, NOT congruent.

According to the Side-Angle-Side (SAS) Theorem for similar triangles, if two sides of one triangle are proportional to two corresponding sides of another triangle, and the angle between these two sides of known measure is the same for each triangle, then these triangles are similar triangles.

We see that the side of measure 4 and the side of measure 8 are proportional because 8 is simply 4x2.  The same can be seen with the sides of measures 6 and 12, 12 is simply 6x2.  Then the angle between each of these sides on either triangle is the same.  So by SAS Theorem for similar triangles, these two triangles are the same.

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THE BIG IDEA

Other than congruence... there is no bigger idea than similarity. Ratios in diagrams constantly show up and at the heart of those most often is the hidden similarity relationship. Similarity shows up in all areas from this point forward - partitioning a segment, angle values in a circle, parallel lines, mid-segments, etc... the list just keep going on and on. Recognizing and using the concepts of similarity is essential for success in geometry.

TRAPS & PITFALLS

When we did proof of congruence we looked for 'hidden' items in the diagram that weren't specifically mentioned in the given such as vertical angles, a common side, angles formed by parallel lines, etc... These same items help us succeed in proof of similarity. The one that is most 'hidden' to the student eye is the common angle. A very common setup for similar triangles is to have an overlapping angle. So in the diagram below ∠A is common in both triangles - this is often missed by students.

PAST CONNECTIONS

In G.CO.A.8 we looked at the criteria for triangle congruence, and now we are doing the same thing but with establishing similarity between figures. Much of the same logic and work takes place for both of these processes.

FUTURE CONNECTIONS

As mentioned in the big idea, establishing figures to be similar is a very common theme throughout the rest of the year. Similarity of triangles shows up very very often because the only required knowledge is two corresponding congruent angles, and this happens in many different cases, it happens almost automatically anytime parallel lines are present in the diagram.

MY REFLECTIONS (over line l )

Nothing really hidden in this objective, we have been teaching similar proof and specifically AA from the beginning of time. I found the students to picked it up quickly because how 'similar' it is to what we did with congruence.

Most proofs of this type are AA and not SAS or SSS even through they are viable ways to also establish similarity. I think they are not used as much because to get the information into the diagram it is difficult, it would probably require giving lengths and I can't recollect ever seeing that. Usually the given expressly gives that the ratios exist which takes any adventure out of it.

Geometry: Common Core (15th Edition)

By charles, randall i., chapter 8 - right triangles and trigonometry - 8-6 law of cosines - practice and problem-solving exercises - page 530: 7, work step by step, update this answer.

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