Free math worksheets

- Look for which sides carry the same tick style. - Translate the markings into a statement about all three side lengths. - Choose the most specific side-length category supported by that information.

1. The same tick mark appears on all three sides, so all three sides are congruent. 2. A triangle with three congruent sides is equilateral. 3. Under the inclusive definition, it is also isosceles, but equilateral is the more specific side-length classification.

Equilateral; all three sides have matching congruence marks.

- Interpret the symbol drawn at the marked angle. - Decide which angle-based triangle category is determined by that symbol. - Side lengths are not needed for this classification.

1. The angle mark shows that one angle measures \(90^\circ\). 2. A triangle with one right angle is a right triangle.

Right triangle

- Compare the three side lengths in each triangle. - How many congruent sides identify an isosceles triangle? - Why is “most specific” important when all three sides are congruent?

1. For a), all three side lengths are different, so the triangle is scalene. 2. For b), all three sides have length \(6\,\text{cm}\). The triangle is equilateral. 3. For c), two sides have the same length, \(a = c = 4\,\text{cm}\), while the third side has a different length. The triangle is isosceles.

a) Scalene, because all three side lengths are different. b) Equilateral, because all three sides are congruent. c) Isosceles, because two sides have length \(4\,\text{cm}\).

- Classify each triangle independently; do not use elimination from the other two. - A single \(90^\circ\) angle determines one angle classification, and a single angle greater than \(90^\circ\) determines another. - To call a triangle acute, check that every marked angle in that triangle is less than \(90^\circ\).

1. In \(\triangle EGF\), the angle at \(G\) is marked as a right angle, so \(\triangle EGF\) is right. 2. In \(\triangle EGH\), the angle at \(H\) is marked \(110^\circ\), so \(\triangle EGH\) is obtuse. 3. In \(\triangle FHG\), the marked angles are \(65^\circ\), \(70^\circ\), and \(45^\circ\). All three are less than \(90^\circ\), so \(\triangle FHG\) is acute.

\(\triangle EGF\): right \(\triangle EGH\): obtuse \(\triangle FHG\): acute

- Identify exactly which sides share a tick style. - Ask what side classification is guaranteed by that marked congruent pair. - For part c), distinguish “not shown” from “shown to be different.”

1. The matching tick marks show that \(\overline{AC}\) and \(\overline{BC}\) are congruent. 2. At least two congruent sides guarantee that the triangle is isosceles. 3. No marking states that \(\overline{AB}\) is congruent to those two sides, so equilateral is not guaranteed.

a) Isosceles b) \(\overline{AC}\) and \(\overline{BC}\) c) No. The markings do not show that all three sides are congruent.

- Compare each angle measure with \(90^\circ\). - One special angle can determine the angle-based class. - Use the name associated with an angle equal to \(90^\circ\).

1. One angle measures exactly \(90^\circ\). 2. A triangle with one \(90^\circ\) angle is a right triangle.

Right triangle; the \(90^\circ\) angle determines the classification.

- Compare the three side lengths with one another. - Count how many pairs of equal side lengths there are. - Match that pattern to a side-based triangle name.

1. The three side lengths are all different. 2. A triangle with three different side lengths is scalene.

Scalene

- Compare the three side lengths. - Look for a repeated length. - Use the side-based name determined by that repetition.

1. Two side lengths are equal: \(7\,\text{in.}\) and \(7\,\text{in.}\). 2. A triangle with at least two congruent sides is isosceles.

Isosceles; the two \(7\,\text{in.}\) sides are congruent.

- Compare every angle with \(90^\circ\). - Check whether any angle is right or obtuse. - Use the category that describes all three angles.

1. Each angle is less than \(90^\circ\). 2. A triangle with three acute angles is an acute triangle.

Acute triangle

- Look for an angle greater than \(90^\circ\). - One obtuse angle is enough to determine the angle-based class. - The other two angles do not need to be equal.

1. One angle measures \(110^\circ\), which is greater than \(90^\circ\). 2. A triangle with one obtuse angle is an obtuse triangle.

Obtuse triangle

- Identify which two sides carry matching tick marks. - Identify the special angle marking separately. - Give the evidence with each classification before combining them.

1. The matching side ticks show two congruent sides, so the triangle is isosceles by side lengths. 2. The angle mark shows one angle is \(90^\circ\), so the triangle is right by angles. 3. The combined classification is an isosceles right triangle.

Isosceles by sides because two sides have matching congruence ticks; right by angles because one angle has a right-angle mark. The combined classification is isosceles right.

- In each panel, identify the matching side ticks and state what they guarantee. - Then identify the exact angle mark that determines the angle classification. - Compare the two classifications only after citing the evidence in both panels.

1. In both figures, two sides have matching tick marks, so both triangles are isosceles. 2. Figure a) has a right-angle mark, so it is a right triangle. 3. Figure b) has an angle marked \(120^\circ\), so it is an obtuse triangle. 4. The triangles share the side classification isosceles but differ as right and obtuse by angles.

a) Both are isosceles; each panel has a marked pair of congruent sides. b) Figure a) is right because it has a right-angle mark; figure b) is obtuse because it has a marked \(120^\circ\) angle. c) Same side class: isosceles. Different angle classes: right and obtuse.

- Read all three degree labels before deciding the angle classification. - Use the side-length statement independently from the angle markings. - An acute classification requires every angle to be less than \(90^\circ\).

1. The marked angles are \(50^\circ\), \(60^\circ\), and \(70^\circ\), so the largest is \(70^\circ\). 2. Because all three side lengths are different, the triangle is scalene. 3. Every marked angle is less than \(90^\circ\), so the triangle is acute. 4. The two-way classification is scalene acute.

a) \(70^\circ\) b) Scalene acute; all three sides have different lengths, and all three marked angles are less than \(90^\circ\).

- Use the side information and the angle information as two separate classifications. - Ask whether any two side lengths are congruent. - Compare the given angle with a right angle.

1. Because all three side lengths are different, the triangle is scalene. 2. Because one angle measures \(100^\circ\), the triangle has an angle greater than \(90^\circ\), so it is obtuse. 3. The two-way classification is scalene obtuse.

scalene obtuse

- Compare all three side lengths, not just the matching pair. - Use the stated angle type independently from the side lengths. - Equilateral requires all three side lengths to be congruent.

1. Exactly two of the given side lengths are equal, so the triangle is isosceles. 2. Every angle is stated to be acute, so its angle classification is acute. 3. The side lengths are not all equal because the third side is \(7\,\text{cm}\), so the triangle is not equilateral. 4. Its two-way classification is isosceles acute.

a) Isosceles b) Acute c) It is not equilateral because the side lengths are not all congruent: \(5\,\text{cm}\), \(5\,\text{cm}\), and \(7\,\text{cm}\).

- Decide the side classification from the stated side relationship before using any angle information. - For the side classification, compare how specific the information “two congruent sides” and “three congruent sides” is. - Classify the angles independently, then combine the two descriptions.

1. Triangle A is guaranteed to be isosceles because at least two sides are congruent. Its \(100^\circ\) angle is obtuse, so its angle classification is obtuse. 2. Triangle B has three different side lengths, so it is scalene. Its right angle makes its angle classification right. 3. Triangle C has three congruent sides, so its most specific side-length classification is equilateral. Its three acute angles make its angle classification acute.

a) Isosceles obtuse b) Scalene right c) Equilateral acute

- Read and report the side labels or congruence marks in each panel. - Choose the most specific side category that those displayed facts guarantee. - Report the right-angle or degree marking that determines each angle classification.

1. Figure a) has side lengths \(3\), \(4\), and \(5\), so its most specific side classification is scalene. It has a right-angle mark, so it is right. 2. Figure b) has a marked pair of congruent sides, so isosceles is guaranteed. It has a \(120^\circ\) angle, so it is obtuse. 3. Figure c) has all three sides marked congruent, so its most specific side classification is equilateral. Its three angles are marked \(60^\circ\), so it is acute.

a) Scalene right: the side labels are \(3\), \(4\), and \(5\), and the diagram has a right-angle mark. b) Isosceles obtuse: two sides have matching congruence ticks, and one angle is marked \(120^\circ\). c) Equilateral acute: all three sides have matching congruence ticks, and the three angles are marked \(60^\circ\).

- Count exactly how many side lengths are equal. - Compare that count with the definition of equilateral. - Then choose the side-based category that matches the given lengths.

1. The triangle has only two equal side lengths, not three. 2. Equilateral means all three sides are congruent. 3. With two congruent sides, the triangle is isosceles.

Riley confused “some equal sides” with “all equal sides.” The triangle is isosceles, not equilateral.

- Check all three angles, not just the majority of them. - Look for an angle equal to or greater than \(90^\circ\). - One special angle can determine the triangle's angle classification.

1. Two angles are acute, but the third angle is exactly \(90^\circ\). 2. A triangle with one right angle is classified as a right triangle. 3. Therefore, Morgan's classification is incorrect.

No. It is a right triangle because it has a \(90^\circ\) angle.

- Use the congruent-side information for the side classification. - Compare \(105^\circ\) with \(90^\circ\) for the angle classification. - A counterexample must satisfy the condition in the claim while disproving its conclusion.

1. Two congruent sides make the triangle isosceles. 2. An angle of \(105^\circ\) is obtuse, so the triangle is obtuse. 3. The triangle is therefore isosceles obtuse, which is a counterexample to Avery's claim.

It is a counterexample. The triangle is isosceles obtuse.

- Work backward from the two words in the required classification. - Ask what diagram marking guarantees the angle-based part. - Ask what separate marking guarantees the side-based part.

1. Mark one angle as a right angle. 2. Put matching congruence tick marks on the two sides that form that right angle. 3. The right-angle mark guarantees the triangle is right, and the matching side marks guarantee it is isosceles.

Add a right-angle mark at one vertex and matching tick marks on the two sides that meet at that vertex.

- Use the side marks for one classification. - Read the labeled angle and compare it with \(90^\circ\) for the other classification. - Combine the two independently justified descriptions.

1. The two matching side ticks show two congruent sides, so the triangle is isosceles. 2. The marked \(110^\circ\) angle is greater than \(90^\circ\), so the triangle is obtuse. 3. The combined classification is isosceles obtuse.

Isosceles obtuse

- Classify the sides of each triangle independently from the given right-angle information. - Compare the number of congruent side lengths in \(R\) and \(S\). - Use the two examples to decide what the word “right” does and does not tell you.

1. Triangle \(R\) has two congruent sides, so it is isosceles. It is given to be right, so it is isosceles right. 2. Triangle \(S\) has three different side lengths, so it is scalene. It is given to be right, so it is scalene right. 3. The two triangles share the angle classification right but have different side classifications. Therefore, the angle classification alone does not determine the side classification.

\(R\): isosceles right \(S\): scalene right A right triangle can have different side classifications.

- Count grid cells along the two sides that meet at \(A\). - Compare a horizontal grid direction with a vertical grid direction. - Use the side result and angle result as two separate classifications before combining them.

1. Segment \(\overline{AB}\) spans \(4\) vertical grid cells, so \(AB=4\) units. Segment \(\overline{AC}\) spans \(4\) horizontal grid cells, so \(AC=4\) units. 2. At \(A\), one side follows a vertical grid line and the other follows a horizontal grid line. These directions form a right angle. 3. Because \(AB=AC\), the triangle is isosceles. Because \(\angle A\) is right, the triangle is right. It is an isosceles right triangle.

a) \(AB=4\) units and \(AC=4\) units b) Right c) Isosceles right

- Check the two sides that meet at the marked angle in each panel. - Use tick marks or length labels to decide whether those two sides are congruent. - A qualifying triangle must satisfy both the side condition and the angle condition.

1. In frame a), the matching side ticks show that \(KL=LM\), and the angle mark at \(L\) shows a right angle. Therefore, \(\triangle KLM\) is isosceles right. 2. In frame b), the angle at \(Q\) is right, but the labeled sides meeting there are \(6\,\text{cm}\) and \(3\,\text{cm}\), so \(\triangle PQR\) is not isosceles. 3. Therefore, \(\triangle KLM\) is the triangular half guaranteed to meet the requirement.

\(\triangle KLM\) in frame a)

- Focus on the words “at least two” in the definition of isosceles. - Ask whether having three congruent sides satisfies a requirement of having two or more congruent sides. - For the most specific name, choose the category that gives the strongest side condition.

1. The phrase “at least two congruent sides” includes triangles with exactly two congruent sides and triangles with three congruent sides. 2. An equilateral triangle has three congruent sides, so it satisfies the definition of isosceles as well as the definition of equilateral. 3. It is not scalene because scalene triangles have no congruent sides. 4. Equilateral is the most specific side-based name because it states that all three sides are congruent.

a) No. Three congruent sides still means at least two congruent sides. b) isosceles and equilateral c) equilateral