5 1 Parallelograms
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1) A parallelogram is a quadrilateral with both pairs of opposite sides parallel. 2) The document proves that in a parallelogram, opposite sides are congruent and opposite angles are congruent. 3) It also states that the diagonals of a parallelogram bisect each other.
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5 1 Parallelograms
1) A parallelogram is a quadrilateral with both pairs of opposite sides parallel.
2) Theorems about parallelograms are that opposite sides are congruent, opposite angles are congruent, and the diagonals bisect each other.
3) Properties of parallelograms include having parallel and congruent opposite sides, congruent opposite angles, and diagonals that bisect each other.
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2. Angle-Angle-Side (AAS) postulate - If two angles and one non-included side of one triangle are congruent to two angles and the non-included side of another triangle, then the triangles are congruent.
The document provides examples of applying these postulates to determine if pairs of triangles are congruent. It also notes there are no AAA or SSA postulates to prove triangles congruent.
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This document provides information about naming and comparing polygons, identifying corresponding parts of polygons, and determining if polygons are congruent. It defines what makes polygons congruent as having corresponding angles and sides that are congruent. It also introduces the Polygon Congruence Postulate and Third Angle Theorem, which can be used to prove polygons or triangles are congruent.
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1. Propositional logic (L1) has a simple syntax consisting of sentence letters like P and Q, and logical connectives like negation (¬), conjunction (∧), and disjunction (∨).
2. The semantics of L1 specifies the truth conditions of sentences based on the truth values (True/False) assigned to basic sentences under an interpretation.
3. The meanings of the logical connectives are given by their truth tables, which determine the truth value of complex sentences based on the truth values of their components. For example, a conjunction is only true if both conjuncts are true.
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* Write and interpret congruence statements
* Use properties of congruent triangles
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1) The student can identify and prove congruent triangles using three criteria: side-side-side (SSS), side-angle-side (SAS), and hypotenuse-leg (HL).
2) For SSS, if three sides of one triangle are congruent to three sides of another, the triangles are congruent.
3) For SAS, if two sides and the included angle of one triangle are congruent to those of another, the triangles are congruent.
4) For HL, if the hypotenuse and one leg are congruent between two right triangles, the triangles are congruent.
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5 1 Parallelograms
1) A parallelogram is a quadrilateral with both pairs of opposite sides parallel.
2) Theorems about parallelograms are that opposite sides are congruent, opposite angles are congruent, and the diagonals bisect each other.
3) Properties of parallelograms include having parallel and congruent opposite sides, congruent opposite angles, and diagonals that bisect each other.
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The document provides examples of applying these postulates to determine if pairs of triangles are congruent. It also notes there are no AAA or SSA postulates to prove triangles congruent.
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This document discusses identifying congruent polygons and triangles. It defines congruent polygons as having the same size and shape, with corresponding angles and sides being congruent. It provides examples of naming corresponding parts of congruent polygons and using properties of congruency to solve problems. Special rules for congruent triangles are given, including side-side-side, side-angle-side, and angle-side-angle. Students are directed to an online resource and practice problems are assigned.
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1) The student can identify and prove congruent triangles using three criteria: side-side-side (SSS), side-angle-side (SAS), and hypotenuse-leg (HL).
2) For SSS, if three sides of one triangle are congruent to three sides of another, the triangles are congruent.
3) For SAS, if two sides and the included angle of one triangle are congruent to those of another, the triangles are congruent.
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The document discusses four right triangle congruence theorems: leg-leg (LL), hypotenuse-angle (HA), leg-angle (LA), and hypotenuse-leg (HL). It then provides three examples of proving right triangles congruent using these theorems, showing the steps of the proofs.
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The document discusses properties of parallelograms including: opposite sides are congruent, opposite angles are congruent, and consecutive angles are supplementary. It also states that the diagonals of a parallelogram bisect each other. Examples are provided to demonstrate using these properties to find lengths, angle measures, and prove statements about parallelograms.
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The student is able to use angles formed by a transversal to show that lines are parallel. Specifically, the student can use corresponding angles, alternate interior angles, alternate exterior angles, and consecutive interior angles to prove lines are parallel if the angles are congruent or supplementary. For example, if the consecutive interior angles are supplementary, then the lines are parallel.
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The document discusses key concepts relating to slope and lines including:
- Two lines are parallel if they have the same slope
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- It provides an example of finding the value of a given four points and asks to determine a such that the lines are perpendicular
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- Side-Side-Side (SSS): If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.
- Side-Angle-Side (SAS): If two sides and the included angle of one triangle are congruent to those of another triangle, then the triangles are congruent.
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The document discusses Euclid's fifth postulate, also known as the parallel postulate. It states that if a straight line falling on two straight lines makes the alternate angles equal to one another, the straight lines will be parallel to one another. Mathematicians were skeptical of this postulate and sought to prove it as a consequence of the other postulates, but were unsuccessful. Alternative statements that are equivalent to the fifth postulate were developed, including Playfair's axiom and the angle-sum of a triangle being equal to two right angles. Attempts to prove the fifth postulate led to insights into its true nature and significance.
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The document provides information about congruent triangles:
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- Examples are provided to demonstrate using properties of congruent triangles to find missing angle measures and prove triangles are congruent by showing corresponding parts are equal.
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This document discusses linear angles formed by parallel lines intersected by a transversal. It begins with an assessment day and learning outcome about using deductive reasoning to justify geometric properties. Students are then introduced to linear angles formed by parallel lines and asked to find measures of angles if one is given. The rest of the document provides tools and definitions for angle relationships formed by parallel lines, such as corresponding angles being congruent. Students practice using these tools and definitions to prove properties about angle measures in different scenarios. The document concludes with instructions to prepare for a quiz by summarizing the mathematics and checking understanding.
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The document discusses definitions, biconditional statements, and their use in geometry. It provides examples of:
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The document discusses relationships between the areas of plane figures such as parallelograms, triangles, and rectangles that share the same base or lie between the same parallels. It defines what it means for figures to be on the same base and between the same parallels. The key points are that parallelograms on the same base and between the same parallels have equal areas, and triangles on the same base and between the same parallels also have equal areas. Formulas for calculating the areas of parallelograms and triangles are reviewed.
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This document discusses different ways to prove that a quadrilateral is a parallelogram, including: if both pairs of opposite sides are congruent; if an angle is supplementary to its consecutive angles; if both pairs of opposite angles are congruent; if the diagonals of a quadrilateral bisect each other; or if one pair of opposite sides is both congruent and parallel. It provides examples of each theorem and assigns practice problems for students to complete.
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Math 150 fall 2020 homework 1 due date friday, october 15,
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- Rectangles and parallelograms both have two sets of parallel sides but rectangles have four right angles while parallelograms have two acute and two obtuse angles.
- Trapezoids have two sides that are parallel and two sides that are not parallel.
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