Triangles: Congruence, Similarity, and Trigonometric Ratios

What is the sum of the angles in a triangle?

180°

What are congruent triangles?

Triangles that have exactly the same length sides and the same size angles.

What are similar triangles?

Triangles that have corresponding equal angles and sides in the same proportion, but are not the same size.

What does the symbol '≅' represent?

It means 'congruent to'.

What is a scale ratio?

The ratio of the lengths of lines on a drawing to the lengths of the objects in the real world.

How can scale be presented on a map?

It can be presented as a statement (e.g., 1,000 km = 1 cm) or as a ratio (e.g., 1:100,000,000).

What is the scale ratio for a drawing of the CN Tower to the actual CN Tower?

1:11,000

What is the scale ratio for house plans to an actual house?

1:200

What is the condition for Side-Side-Side (SSS) congruency?

All three corresponding sides of the triangles are equal.

What is the condition for Angle-Side-Angle (ASA) congruency?

Two angles and the side between them in one triangle are equal to the corresponding angles and side in another triangle.

What is the condition for Side-Angle-Side (SAS) congruency?

Two sides and the angle between them in one triangle are equal to the corresponding sides and angle in another triangle.

If a triangle has angles measuring 30°, 60°, and 90°, what can be said about its angles?

The angles add up to 180°.

What is the relationship between the longest side of a triangle and its angles?

The longest side is opposite the largest angle.

What is the relationship between the shortest side of a triangle and its angles?

The shortest side is opposite the smallest angle.

How do you write a congruency statement for triangles?

You must match the equal angles and sides in the same order.

What is a scale model?

A representation of an object that maintains the same proportions as the actual object.

What is the scale ratio for a model of a sports car to the actual sports car?

1:60

What is the scale ratio for a diagram of a human cell to an actual human cell?

100:1

What is the scale ratio for a drawing of an insect to an actual insect?

4.5:1

What is the scale ratio for the distance measured on a map to the actual distance on the earth's surface?

1:100,000,000

What is the best scale to compare the width and height of a TV screen measuring 1.25 m wide and 70 cm high?

The best scale would be 1.25 m : 0.70 m.

What is the purpose of using triangles in problem-solving?

Triangles provide a useful mathematical tool for various applications in fields like carpentry, engineering, and design.

What is the significance of scale in geometry?

Scale helps relate the dimensions of objects in drawings to their actual sizes in the real world.

What do you need to verify to show that triangles are congruent?

You need to determine if they satisfy one of the congruency conditions: SSS, SAS, or ASA.

What is the relationship between corresponding angles in similar triangles?

All corresponding angles are equal.

What is the scale ratio for the dimensions of an ice hockey rink measuring 60 m long by 30 m wide, represented in a drawing of 30 cm long?

60 m : 30 cm

What does the symbol '∼' represent in geometry?

It means 'similar to.'

What is the importance of the order in a similarity statement?

The equal angles in both triangles must be written in the same order.

What must be true about the corresponding angles of similar triangles?

The corresponding angles are always equal.

What must be true about the corresponding sides of similar triangles?

The ratio of all corresponding sides is always the same.

What are the three conditions to identify similar triangles?

Angle-angle similarity (AA), Side-side-side similarity (SSS), Side-angle-side similarity (SAS).

What does angle-angle similarity (AA) indicate?

If two pairs of angles are equal, then the triangles are similar.

What is the relationship between congruent and similar triangles?

Congruent triangles have equal angles and sides; similar triangles have equal angles but proportional sides.

How can you determine the height of an object using similar triangles?

By setting up a proportion based on the corresponding sides of the similar triangles.

What is the significance of the angle of incidence and reflection in similar triangles?

They help establish that the triangles formed are similar due to equal angles.

In the example of a flagpole and a meter stick, what is the height of the flagpole if its shadow is 14 m and the meter stick's shadow is 2 m?

The height of the flagpole is 7 m.

How do you find the height of the inukshuk monument using similar triangles?

By setting up a proportion based on the distances from the observer to the mirror and the monument.

What is the height of the inukshuk monument if the distance from Cornella to the mirror is 0.5 m and from her eyes to the ground is 1.6 m?

The height of the inukshuk monument is 3.2 m.

What must be true for triangles to be considered congruent?

Three corresponding angles and three corresponding sides must be equal.

What is the definition of similar triangles?

Triangles that have the same shape but not necessarily the same size, with corresponding angles equal and sides in proportion.

What is the angle sum property of triangles?

The angles in a triangle must add up to 180°.

What is the purpose of using a graphing app in exploring similar triangles?

To observe how changing the shape of triangles affects their angles and side ratios.

What is the first step in solving problems involving similar triangles?

Identify the similar triangles and their corresponding sides.

What is the role of a similarity statement in triangle similarity?

It shows which angles are equal and which sides correspond proportionally.

What is the relationship between the angles of incidence and reflection?

They are equal, which helps establish the similarity of triangles formed by light reflection.

What is an example of a real-world application of similar triangles?

Using similar triangles to measure the height of tall objects like trees or buildings.

What does the term 'proportional sides' mean in the context of similar triangles?

It refers to the sides of similar triangles that maintain a constant ratio to each other.

What is the significance of the 90° angle in determining triangle similarity?

It helps confirm that the triangles are similar when combined with other equal angles.

What is the conclusion drawn from exploring similar triangles?

Understanding the concepts of similarity and congruence is essential for solving geometric problems.

What is the hypotenuse in a right triangle?

The hypotenuse is the side across from the 90° angle.

What is the opposite side in a right triangle?

The opposite side is the side across from the reference angle.

What is the adjacent side in a right triangle?

The adjacent side is the side between the reference angle and the 90° angle.

What does the acronym SOH-CAH-TOA represent?

SOH-CAH-TOA helps recall the definitions of sine, cosine, and tangent in relation to the sides of a right triangle.

What is the sine ratio?

The sine ratio is the ratio of the length of the opposite side to the length of the hypotenuse.

What is the cosine ratio?

The cosine ratio is the ratio of the length of the adjacent side to the length of the hypotenuse.

What is the tangent ratio?

The tangent ratio is the ratio of the length of the opposite side to the length of the adjacent side.

How do corresponding angles behave in similar triangles?

Corresponding angles in similar triangles are equal.

How do corresponding sides behave in similar triangles?

All pairs of corresponding sides of similar triangles are in the same ratio.

What happens to the primary trigonometric ratios when the size of a right triangle changes?

The primary trigonometric ratios remain the same for a given reference angle, regardless of the size of the triangle.

What is the significance of the reference angle in trigonometry?

The reference angle determines the values of the primary trigonometric ratios.

What is the relationship between similar triangles and trigonometric ratios?

The primary trigonometric ratios apply to right triangles, which can be analyzed using the properties of similar triangles.

What is the formula for sine in a right triangle?

Sine = Opposite side / Hypotenuse.

What is the formula for cosine in a right triangle?

Cosine = Adjacent side / Hypotenuse.

What is the formula for tangent in a right triangle?

Tangent = Opposite side / Adjacent side.

What is the first step in determining if two triangles are similar?

Check if their corresponding angles are equal.

What is the second step in determining if two triangles are similar?

Verify that the ratios of their corresponding sides are equal.

What is the importance of labeling sides in a right triangle?

Consistent labeling helps apply properties correctly and communicate ideas clearly.

What is the role of Greek mathematicians in the development of trigonometry?

Greek mathematicians used triangles to solve problems in astronomy and navigation, contributing to the field of trigonometry.

What does the term 'trigonometry' mean?

Trigonometry comes from Greek words meaning 'three,' 'angle,' and 'measurement.'

What is the significance of the 90° angle in a right triangle?

The 90° angle is essential for defining the hypotenuse and the relationships between the sides.

What is the relationship between the primary trigonometric ratios and the size of the triangle?

The primary trigonometric ratios are consistent for a given reference angle, regardless of triangle size.

What should you do if you need to find the trigonometric ratios using a calculator?

Activate the degree mode on your calculator before calculating trigonometric ratios.

What is the purpose of the learning activity described?

To explore the properties of triangles and understand the primary trigonometric ratios.

What key function is used to find angles in trigonometry?

The inverse trigonometric functions, such as [SIN−1], [COS−1], or [TAN−1].

What does the acronym 'SOH-CAH-TOA' represent?

It helps recall the definitions of sine, cosine, and tangent in relation to the sides of a right triangle.

What is the primary purpose of trigonometric ratios?

To solve for unknown sides and angles in right triangles.

What is the sine law?

A mathematical rule stating that the ratios of the lengths of sides of a triangle to the sines of their opposite angles are equal.

What is the formula for the sine law?

a/sin(A) = b/sin(B) = c/sin(C), where a, b, and c are side lengths and A, B, and C are the opposite angles.

What angle is formed where a ladder meets the ground if it is 5 m long and touches a wall 4 m above the ground?

Use trigonometric ratios to calculate the angle.

How can you find the distance from the base of a wall to the bottom of a ladder?

By using the cosine ratio or the Pythagorean theorem.

What is the angle of elevation to the top of a flagpole if the distance from the flagpole is 20 m and the angle is 54°?

Use the tangent ratio to find the height of the flagpole.

What is the significance of the [DEG] key on a calculator?

It sets the calculator to degree mode for trigonometric calculations.

What should the angle be for a ladder to be considered safe when meeting the ground?

Between 65° and 80°.

What historical figure is known for calculating trajectories by hand for NASA?

Mathematician Katherine Johnson.

What is the Pythagorean theorem used for?

To solve for unknown sides in right triangles.

What does the term 'acute triangle' refer to?

A triangle in which all three angles are less than 90°.

How can you check if your calculator is set correctly for trigonometric calculations?

Ensure the correct mode (degree or radian) is selected before performing calculations.

What is the first step to solve for an unknown angle using trigonometric ratios?

Identify the known sides and use the appropriate trigonometric function.

What is the formula for calculating the sine of an angle?

sin(θ) = opposite side / hypotenuse.

What does the term 'inverse trigonometric functions' refer to?

Functions that allow you to find the angle when the ratio is known.

What is the importance of rounding answers in trigonometric calculations?

To provide a practical and usable answer, typically to the nearest degree.

What is the relationship between the sides and angles in a right triangle?

The sides are proportional to the sine of their opposite angles.

What is the role of a scientific calculator in trigonometry?

To perform calculations involving trigonometric ratios and angles.

What should you do if your answer does not match the expected result in trigonometric calculations?

Check your calculations and ensure the correct mode is set on your calculator.

What is the significance of the angle θ in trigonometry?

It represents the angle being calculated or referenced in trigonometric functions.

What is the purpose of the portfolio item submission mentioned in the notes?

To evaluate and check your understanding of the trigonometric concepts learned.

What is the expected output when using the sine law in calculations?

To find unknown angles or side lengths in non-right triangles.

What is required to use the sine law?

You must know either two sides and one angle across from a known side, or two angles and any side.

How can the sine law be expressed when solving for an angle?

Using the formula: a/sin(A) = b/sin(B) = c/sin(C).

How can the sine law be expressed when solving for a side length?

Using the formula: a/sin(A) = b/sin(B) = c/sin(C).